Derniers messages sur Zeste de Savoirhttps://zestedesavoir.com/forums/2019-03-05T21:42:14+01:00Les derniers messages parus sur le forum de Zeste de Savoir.Résolution numérique d’une équation aux dérivées partielles, message #1999902019-03-05T21:42:14+01:00Tintinfil/@Tintinfilhttps://zestedesavoir.com/forums/sujet/12143/resolution-numerique-dune-equation-aux-derivees-partielles/?page=1#p199990<p>J’ai réussi à implémenter le schéma explicite et les résultats ne sont pas trop mal. Je vais regarder plus en détail le problème physique pour trouver de meilleures conditions aux bords (mon prof me conseille les flux constants aux bords). Je vais voir aussi pour le schéma implicite, mais le calcule de la jacobienne me paraît lourd. Du coup, problème résolu ! Merci <a href="/membres/voir/adri1/" rel="nofollow" class="ping ping-link">@<span class="ping-username">adri1</span></a> d’avoir pris le temps de m’aider !!!</p>Résolution numérique d’une équation aux dérivées partielles, message #1999692019-03-05T13:46:09+01:00adri1/@adri1https://zestedesavoir.com/forums/sujet/12143/resolution-numerique-dune-equation-aux-derivees-partielles/?page=1#p199969<p>Ton résultat ne me parait pas bon pour une raison simple : tu donnes le même poids à <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>h</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">h_{k-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.902771em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span></span> à <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>h</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">h_{k+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.902771em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span></span> alors que la situation est asymétrique.</p>Résolution numérique d’une équation aux dérivées partielles, message #1999532019-03-05T08:47:42+01:00Tintinfil/@Tintinfilhttps://zestedesavoir.com/forums/sujet/12143/resolution-numerique-dune-equation-aux-derivees-partielles/?page=1#p199953<p>Au temps pour moi, on obtient pas le bon résultat pour <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>ε</mi><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\varepsilon = -\frac12\Delta x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">ε</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord">Δ</span><span class="mord mathdefault">x</span></span></span></span></span>. Voici donc la relation que j’obtiens :
<span class="inlineMath inlineMathDouble"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="normal">∂</mi><mi>x</mi></msub><mi>h</mi><mo>(</mo><msub><mi>x</mi><mi>i</mi></msub><mo>−</mo><mi mathvariant="normal">ℓ</mi><mo separator="true">,</mo><mi>t</mi><mo>)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>δ</mi></mrow></mfrac><mrow><mo fence="true">(</mo><mi>h</mi><mo>(</mo><msub><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo separator="true">,</mo><mi>t</mi><mo>)</mo><mo>−</mo><mfrac><mrow><mn>2</mn><mi>ε</mi></mrow><mi>δ</mi></mfrac><mo>[</mo><mi>h</mi><mo>(</mo><msub><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo separator="true">,</mo><mi>t</mi><mo>)</mo><mo>−</mo><mn>2</mn><mi>h</mi><mo>(</mo><msub><mi>x</mi><mi>k</mi></msub><mo separator="true">,</mo><mi>t</mi><mo>)</mo><mo>+</mo><mi>h</mi><mo>(</mo><msub><mi>x</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo separator="true">,</mo><mi>t</mi><mo>)</mo><mo>]</mo><mo>−</mo><mi>h</mi><mo>(</mo><msub><mi>x</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo separator="true">,</mo><mi>t</mi><mo>)</mo><mo fence="true">)</mo></mrow><mo>+</mo><mi>O</mi><mo>(</mo><mi>δ</mi><mo>)</mo><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">\partial_x h(x_i - \ell, t) = \frac{1}{2\delta}\left(h(x_{k + 1}, t) - \frac{2\varepsilon}{\delta}[h(x_{k + 1}, t) - 2h(x_k, t) + h(x_{k - 1}, t)] - h(x_{k - 1}, t)\right) + O(\delta).</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.05556em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathdefault">h</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">ℓ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.40003em;vertical-align:-0.95003em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord mathdefault">h</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord mathdefault">ε</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">[</span><span class="mord mathdefault">h</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">2</span><span class="mord mathdefault">h</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault">h</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">t</span><span class="mclose">)</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault">h</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">t</span><span class="mclose">)</span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mclose">)</span><span class="mord">.</span></span></span></span></span></span>
(J’ai noté <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>δ</mi><mo>=</mo><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\delta = \Delta x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord">Δ</span><span class="mord mathdefault">x</span></span></span></span></span> pour simplifier.) Je crois que c’est la bonne !</p>Résolution numérique d’une équation aux dérivées partielles, message #1999502019-03-05T01:57:38+01:00adri1/@adri1https://zestedesavoir.com/forums/sujet/12143/resolution-numerique-dune-equation-aux-derivees-partielles/?page=1#p199950<figure><blockquote>
<p>ça marche pour <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>ε</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\varepsilon = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">ε</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span> et <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>ε</mi><mo>=</mo><mo>±</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\varepsilon = \pm\frac12\Delta x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">ε</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord">±</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord">Δ</span><span class="mord mathdefault">x</span></span></span></span></span>.</p>
</blockquote><figcaption><a href="https://zestedesavoir.com/forums/sujet/12143/resolution-numerique-dune-equation-aux-derivees-partielles/?page=1#p199946">Teguad</a></figcaption></figure>
<p>Hmm, ça me parait bizarre que ton expression soit valide à la fois pour <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>ε</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>δ</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\varepsilon=\frac12\delta x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">ε</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord mathdefault">x</span></span></span></span></span> et <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>ε</mi><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>δ</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\varepsilon=-\frac12\delta x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">ε</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord mathdefault">x</span></span></span></span></span>. Ces cas là, bien que correspondant à la même configuration, sont en fait différent en pratique (le nœud <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>x</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">x_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span> sera d’un côté ou de l’autre), c’est pour ça qu’il est important de définir si l’intervalle de valeurs que peut prendre <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>ε</mi></mrow><annotation encoding="application/x-tex">\varepsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">ε</span></span></span></span></span> est ouvert côté <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>δ</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\frac12\delta x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord mathdefault">x</span></span></span></span></span> ou côté <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>δ</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">-\frac12\delta x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord mathdefault">x</span></span></span></span></span>. J’ai du mal à voir comment tu peux obtenir une expression valide sans que l’on ait à se soucier de la borne que l’on autorise comme valeur possible.</p>Résolution numérique d’une équation aux dérivées partielles, message #1999462019-03-04T22:09:29+01:00Tintinfil/@Tintinfilhttps://zestedesavoir.com/forums/sujet/12143/resolution-numerique-dune-equation-aux-derivees-partielles/?page=1#p199946<p>Merci beaucoup ! Je viens de réussir à l’exprimer et je pense que c’est la bonne relation car ça marche pour <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>ε</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\varepsilon = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">ε</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span> et <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>ε</mi><mo>=</mo><mo>±</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\varepsilon = \pm\frac12\Delta x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">ε</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord">±</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord">Δ</span><span class="mord mathdefault">x</span></span></span></span></span>. Je vais essayer d’implémenter ça en Python. Et si ça marche pas, je passerai à la version implicite.</p>Résolution numérique d’une équation aux dérivées partielles, message #1999362019-03-04T20:11:27+01:00adri1/@adri1https://zestedesavoir.com/forums/sujet/12143/resolution-numerique-dune-equation-aux-derivees-partielles/?page=1#p199936<blockquote>
<p>Y’a un moyen simple de l’obtenir ?</p>
</blockquote>
<p>Oui. Je te mets une aide en secret si tu coinces.</p>
<div class="custom-block custom-block-spoiler"><div class="custom-block-body"><p>Calcule <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>h</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>h</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">h_{k+1}-h_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.902771em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span> et <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>h</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>h</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">h_{k-1}-h_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.902771em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span> pour te débarrasser des <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>h</mi><mi mathvariant="normal">ℓ</mi></msub></mrow><annotation encoding="application/x-tex">h_\ell</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">ℓ</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span>. Multiplie l’une des deux équations que tu obtiens par le facteur qui va bien pour te débarrasser des <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><msubsup><mi mathvariant="normal">∂</mi><mi>x</mi><mn>2</mn></msubsup><mi>h</mi><msub><mo>)</mo><mi mathvariant="normal">ℓ</mi></msub></mrow><annotation encoding="application/x-tex">(\partial_x^2h)_\ell</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.05556em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">x</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord mathdefault">h</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">ℓ</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span> par différence. Puis pour être sûr, vérifie que les cas particuliers <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>ε</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\varepsilon=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">ε</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span> et <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>ε</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>δ</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\varepsilon=\frac12\delta x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">ε</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord mathdefault">x</span></span></span></span></span> donnent un résultat cohérent avec le développement que tu as déjà pour <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><msub><mi mathvariant="normal">∂</mi><mi>x</mi></msub><mi>h</mi><msub><mo>)</mo><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">(\partial_xh)_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.05556em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathdefault">h</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span> (i.e. celui où l’on évalue la dérivé première à mi-chemin de deux nœuds connus).</p></div></div>Résolution numérique d’une équation aux dérivées partielles, message #1999082019-03-04T13:31:06+01:00Tintinfil/@Tintinfilhttps://zestedesavoir.com/forums/sujet/12143/resolution-numerique-dune-equation-aux-derivees-partielles/?page=1#p199908<p>Oups désolé pour le temps de réponse. Merci, je vais essayer de me débrouiller avec ça. C’est ce que je voulais faire, en disant que mon <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>x</mi><mrow><mi>j</mi><mo>−</mo><mi>p</mi></mrow></msub></mrow><annotation encoding="application/x-tex">x_{j - p}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span><span class="mbin mtight">−</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></span> était le plus proche de <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>x</mi><mi>j</mi></msub><mo>−</mo><mi mathvariant="normal">ℓ</mi></mrow><annotation encoding="application/x-tex">x_j - \ell</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8694379999999999em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord">ℓ</span></span></span></span></span>. Mais en effet, ça peut être <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>x</mi><mrow><mi>j</mi><mo>−</mo><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">x_{j - p - 1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span><span class="mbin mtight">−</span><span class="mord mathdefault mtight">p</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></span>. Je vais redéfinir mon <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>ε</mi></mrow><annotation encoding="application/x-tex">\varepsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">ε</span></span></span></span></span>.</p>
<p>Ah oui, c’est vrai qu’on a même un <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>O</mi><mo>(</mo><mi mathvariant="normal">Δ</mi><msup><mi>x</mi><mn>2</mn></msup><mo>)</mo></mrow><annotation encoding="application/x-tex">O(\Delta x^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span>, je n’avais pas vu, mais ça ne changera pas grand-chose à mon implémentation. <img src="/static/smileys/heureux.png" alt=":D" class="smiley"></p>
<p><strong>Edit :</strong> Je n’arrive toujours pas à faire des combinaisons linéaires pour obtenir <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="normal">∂</mi><mi>x</mi></msub><mi>h</mi><mo>(</mo><mi>x</mi><mo>−</mo><mi mathvariant="normal">ℓ</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">\partial_x h(x - \ell)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.05556em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathdefault">h</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">ℓ</span><span class="mclose">)</span></span></span></span></span>. Y’a un moyen simple de l’obtenir ? Est-ce qu’il faut négliger des termes à un moment ?</p>
<p><strong>Edit 2 :</strong> Je suis sur une piste. <img src="/static/smileys/smile.png" alt=":)" class="smiley"></p>Résolution numérique d’une équation aux dérivées partielles, message #1998492019-03-03T15:24:42+01:00adri1/@adri1https://zestedesavoir.com/forums/sujet/12143/resolution-numerique-dune-equation-aux-derivees-partielles/?page=1#p199849<p>Salut,</p>
<p>Pour les premiers développements, tu les as écrit au premier ordre, pas au second. Si tu ajoutes les termes d’ordre deux, tu vas voir qu’ils partent en même temps que les termes d’ordre 0 et donc que ton erreur est en fait mieux que <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">o(1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">o</span><span class="mopen">(</span><span class="mord">1</span><span class="mclose">)</span></span></span></span></span>, elle est <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>o</mi><mo>(</mo><mi>δ</mi><mi>x</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">o(\delta x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">o</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord mathdefault">x</span><span class="mclose">)</span></span></span></span></span> (ou comme on le note plus souvent <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>O</mi><mo>(</mo><mi>δ</mi><msup><mi>x</mi><mn>2</mn></msup><mo>)</mo></mrow><annotation encoding="application/x-tex">O(\delta x^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span>, d’où le fait que l’on dise que cette approximation est de deuxième ordre).</p>
<p>Pour la suite, tu te compliques pas mal la vie… Disons que le nœud <em>le plus proche</em> de <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>x</mi><mi mathvariant="normal">ℓ</mi></msub><mo>=</mo><mi>x</mi><mo>−</mo><mi mathvariant="normal">ℓ</mi></mrow><annotation encoding="application/x-tex">x_\ell=x-\ell</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">ℓ</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord">ℓ</span></span></span></span></span> est <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>x</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">x_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span> avec</p>
<p><span class="inlineMath inlineMathDouble"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>x</mi><mi>k</mi></msub><mo>−</mo><msub><mi>x</mi><mi mathvariant="normal">ℓ</mi></msub><mo>=</mo><mi>ε</mi><mo>∈</mo><mrow><mo fence="true">]</mo><mo>−</mo><mfrac><mrow><mi>δ</mi><mi>x</mi></mrow><mn>2</mn></mfrac><mo separator="true">,</mo><mfrac><mrow><mi>δ</mi><mi>x</mi></mrow><mn>2</mn></mfrac><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">x_k-x_\ell=\varepsilon\in\left]-\dfrac{\delta x}{2},\dfrac{\delta x}{2}\right]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">ℓ</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathdefault">ε</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.40003em;vertical-align:-0.95003em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord mathdefault">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord mathdefault">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span></span></span></span></span></span></p>
<p>On a</p>
<p><span class="inlineMath inlineMathDouble"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>h</mi><mi>k</mi></msub><mo>=</mo><msub><mi>h</mi><mi mathvariant="normal">ℓ</mi></msub><mo>+</mo><mi>ε</mi><mo>(</mo><msub><mi mathvariant="normal">∂</mi><mi>x</mi></msub><mi>h</mi><msub><mo>)</mo><mi mathvariant="normal">ℓ</mi></msub><mo>+</mo><mfrac><msup><mi>ε</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>(</mo><msubsup><mi mathvariant="normal">∂</mi><mi>x</mi><mn>2</mn></msubsup><mi>h</mi><msub><mo>)</mo><mi mathvariant="normal">ℓ</mi></msub><mo>+</mo><mi>o</mi><mo>(</mo><mi>δ</mi><msup><mi>x</mi><mn>2</mn></msup><mo>)</mo></mrow><annotation encoding="application/x-tex">h_k = h_\ell + \varepsilon(\partial_x h)_\ell + \frac{\varepsilon^2}{2}(\partial_x^2 h)_\ell+o(\delta x^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">ℓ</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">ε</span><span class="mopen">(</span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.05556em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathdefault">h</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">ℓ</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:2.177108em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">ε</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">(</span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.05556em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">x</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord mathdefault">h</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">ℓ</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.1141079999999999em;vertical-align:-0.25em;"></span><span class="mord mathdefault">o</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span></p>
<p>Puis sur les points de part et d’autre (je te laisse réfléchir à ce qu’il faut faire si <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>x</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">x_{k-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.638891em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span></span> est en dehors du domaine) :</p>
<p><span class="inlineMath inlineMathDouble"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>h</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>h</mi><mi mathvariant="normal">ℓ</mi></msub><mo>+</mo><mo>(</mo><mi>δ</mi><mi>x</mi><mo>+</mo><mi>ε</mi><mo>)</mo><mo>(</mo><msub><mi mathvariant="normal">∂</mi><mi>x</mi></msub><mi>h</mi><msub><mo>)</mo><mi mathvariant="normal">ℓ</mi></msub><mo>+</mo><mfrac><mrow><mo>(</mo><mi>δ</mi><mi>x</mi><mo>+</mo><mi>ε</mi><msup><mo>)</mo><mn>2</mn></msup></mrow><mn>2</mn></mfrac><mo>(</mo><msubsup><mi mathvariant="normal">∂</mi><mi>x</mi><mn>2</mn></msubsup><mi>h</mi><msub><mo>)</mo><mi mathvariant="normal">ℓ</mi></msub><mo>+</mo><mi>o</mi><mo>(</mo><mi>δ</mi><msup><mi>x</mi><mn>2</mn></msup><mo>)</mo></mrow><annotation encoding="application/x-tex">h_{k+1} = h_\ell + (\delta x+\varepsilon)(\partial_xh)_\ell + \frac{(\delta x+\varepsilon)^2}{2}(\partial_x^2 h)_\ell+o(\delta x^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.902771em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">ℓ</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">ε</span><span class="mclose">)</span><span class="mopen">(</span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.05556em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathdefault">h</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">ℓ</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:2.177108em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault">ε</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">(</span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.05556em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">x</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord mathdefault">h</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">ℓ</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.1141079999999999em;vertical-align:-0.25em;"></span><span class="mord mathdefault">o</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span></p>
<p><span class="inlineMath inlineMathDouble"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>h</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>h</mi><mi mathvariant="normal">ℓ</mi></msub><mo>+</mo><mo>(</mo><mi>ε</mi><mo>−</mo><mi>δ</mi><mi>x</mi><mo>)</mo><mo>(</mo><msub><mi mathvariant="normal">∂</mi><mi>x</mi></msub><mi>h</mi><msub><mo>)</mo><mi mathvariant="normal">ℓ</mi></msub><mo>+</mo><mfrac><mrow><mo>(</mo><mi>ε</mi><mo>−</mo><mi>δ</mi><mi>x</mi><msup><mo>)</mo><mn>2</mn></msup></mrow><mn>2</mn></mfrac><mo>(</mo><msubsup><mi mathvariant="normal">∂</mi><mi>x</mi><mn>2</mn></msubsup><mi>h</mi><msub><mo>)</mo><mi mathvariant="normal">ℓ</mi></msub><mo>+</mo><mi>o</mi><mo>(</mo><mi>δ</mi><msup><mi>x</mi><mn>2</mn></msup><mo>)</mo></mrow><annotation encoding="application/x-tex">h_{k-1} = h_\ell + (\varepsilon-\delta x)(\partial_xh)_\ell + \frac{(\varepsilon-\delta x)^2}{2}(\partial_x^2 h)_\ell+o(\delta x^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.902771em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">ℓ</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathdefault">ε</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord mathdefault">x</span><span class="mclose">)</span><span class="mopen">(</span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.05556em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathdefault">h</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">ℓ</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:2.177108em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathdefault">ε</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord mathdefault">x</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">(</span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.05556em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">x</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord mathdefault">h</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">ℓ</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.1141079999999999em;vertical-align:-0.25em;"></span><span class="mord mathdefault">o</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span></p>
<p>Il n’y a plus qu’à se débarrasser des termes <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>h</mi><mi mathvariant="normal">ℓ</mi></msub></mrow><annotation encoding="application/x-tex">h_\ell</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">ℓ</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span> et <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><msubsup><mi mathvariant="normal">∂</mi><mi>x</mi><mn>2</mn></msubsup><mi>h</mi><msub><mo>)</mo><mi mathvariant="normal">ℓ</mi></msub></mrow><annotation encoding="application/x-tex">(\partial_x^2 h)_\ell</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.05556em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">x</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord mathdefault">h</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">ℓ</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span> par combinaison linéaire, je te laisse essayer ça.</p>Résolution numérique d’une équation aux dérivées partielles, message #1998422019-03-03T11:05:25+01:00Tintinfil/@Tintinfilhttps://zestedesavoir.com/forums/sujet/12143/resolution-numerique-dune-equation-aux-derivees-partielles/?page=1#p199842<p>Salut ! J’ai un peu commencé à mettre en place ce que tu m’as dit, mais je bloque au moment de l’approximation de <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="normal">∂</mi><mi>x</mi></msub><mi>h</mi></mrow><annotation encoding="application/x-tex">\partial_x h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.05556em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathdefault">h</span></span></span></span></span> en <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi><mo>−</mo><mi mathvariant="normal">ℓ</mi></mrow><annotation encoding="application/x-tex">x - \ell</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord">ℓ</span></span></span></span></span>. Je n’ai réussi qu’à trouver deux développements de Taylor, mais pas le troisième. En l’état, je n’arrive pas à faire une combinaison linéaire pour me débarrasser de la dérivée seconde. De plus, si on s’arrête au premier degré, j’ai un truc, mais je ne pense pas que ça fonctionne. Je te met ci-dessous ce que j’ai fait. Merci de ton aide !</p>
<div class="custom-block custom-block-spoiler"><div class="custom-block-body"><p><img src="/media/galleries/464/7489078a-b362-4ef2-acf5-1bf6cf3e290b.png"></p></div></div>Résolution numérique d’une équation aux dérivées partielles, message #1994372019-02-23T19:49:25+01:00adri1/@adri1https://zestedesavoir.com/forums/sujet/12143/resolution-numerique-dune-equation-aux-derivees-partielles/?page=1#p199437<p>Tu auras besoin de trois points d’ailleurs, pas deux. Sinon tu vas perdre en précision.</p>Résolution numérique d’une équation aux dérivées partielles, message #1994362019-02-23T19:45:05+01:00Tintinfil/@Tintinfilhttps://zestedesavoir.com/forums/sujet/12143/resolution-numerique-dune-equation-aux-derivees-partielles/?page=1#p199436<p>Merci beaucoup pour ces solutions. Je n’avais pas du tout pensé à circulariser l’espace. Pour ton deuxième point, c’est ce que je voulais un peu faire avec les conditions de Neumann. Je vais essayer de mettre en place les deux solutions et voir laquelle est la meilleure.</p>
<p>Pour ce qui est de l’évaluation de <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="normal">∂</mi><mi>x</mi></msub><mi>h</mi></mrow><annotation encoding="application/x-tex">\partial_x h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.05556em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathdefault">h</span></span></span></span></span>, je vais tenter de faire les DL moi-même et, si je n’y arrive pas, je te demanderai (ce qui sera fort probable). <img src="/static/smileys/langue.png" alt=":p" class="smiley"> Encore merci ! Je te tiens au courant de mes avancées.</p>Résolution numérique d’une équation aux dérivées partielles, message #1994352019-02-23T19:16:03+01:00adri1/@adri1https://zestedesavoir.com/forums/sujet/12143/resolution-numerique-dune-equation-aux-derivees-partielles/?page=1#p199435<p>Pour le <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi><mo>−</mo><mi mathvariant="normal">ℓ</mi></mrow><annotation encoding="application/x-tex">x-\ell</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord">ℓ</span></span></span></span></span> qui sort, c’est pas forcément un problème, tu as deux options simples (et des choses plus subtiles éventuellement, mais je n’ai pas assez réfléchi au problème physique) :</p>
<ul>
<li>rendre ton espace circulaire, si on dit que ton espace est <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo separator="true">,</mo><mi>L</mi><mo>]</mo></mrow><annotation encoding="application/x-tex">x\in[0,L]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">L</span><span class="mclose">]</span></span></span></span></span>, tu peux ensuite le prolonger par périodicité et dire que le point <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>−</mo><mi mathvariant="normal">ℓ</mi></mrow><annotation encoding="application/x-tex">-\ell</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em;"></span><span class="mord">−</span><span class="mord">ℓ</span></span></span></span></span> est le même que le point <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>L</mi><mo>−</mo><mi mathvariant="normal">ℓ</mi></mrow><annotation encoding="application/x-tex">L-\ell</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">L</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord">ℓ</span></span></span></span></span> (et donc que le point <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span> et le point <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault">L</span></span></span></span></span> sont identiques) ;</li>
<li>dire que tout est plat en dehors du domaine et donc avoir <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="normal">∂</mi><mi>x</mi></msub><mi>h</mi></mrow><annotation encoding="application/x-tex">\partial_xh</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.05556em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathdefault">h</span></span></span></span></span> nul à l’extérieur.</li>
</ul>
<p>Ensuite, pour évaluer <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="normal">∂</mi><mi>x</mi></msub><mi>h</mi></mrow><annotation encoding="application/x-tex">\partial_x h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.05556em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathdefault">h</span></span></span></span></span> en un point qui n’est pas forcément un des nœuds de calculs, ce n’est pas un problème non plus. Pour trouver l’expression en fonction de la valeur des deux points qui l’encadrent, écris la valeur de <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault">h</span></span></span></span></span> en ce point en fonction de celle des deux nœuds qui l’encadrent comme un DL au deuxième ordre et débarrasse toi des dérivées secondes par combinaison linéaire. Je peux te guider un peu plus si tu as du mal.</p>Résolution numérique d’une équation aux dérivées partielles, message #1994332019-02-23T18:06:23+01:00Tintinfil/@Tintinfilhttps://zestedesavoir.com/forums/sujet/12143/resolution-numerique-dune-equation-aux-derivees-partielles/?page=1#p199433<p>Merci pour ces conseils ! Pour ton deuxième point, le truc c’est que justement je ne sais pas gérer l’évaluation de <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="normal">∂</mi><mi>x</mi></msub><mi>h</mi></mrow><annotation encoding="application/x-tex">\partial_x h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.05556em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathdefault">h</span></span></span></span></span> au point <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mi>x</mi><mo>−</mo><mi mathvariant="normal">ℓ</mi><mo separator="true">,</mo><mi>t</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">(x - \ell, t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">ℓ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">t</span><span class="mclose">)</span></span></span></span></span>… L’avantage de prendre <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">ℓ</mi><mo>=</mo><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\ell = \Delta x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord">ℓ</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord">Δ</span><span class="mord mathdefault">x</span></span></span></span></span> + Neumann était que je pouvais bien gérer aux bords. Si je prends un <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">ℓ</mi></mrow><annotation encoding="application/x-tex">\ell</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord">ℓ</span></span></span></span></span> quelconque, comme je l’ai dit, pour <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span>, il va falloir que j’évalue en <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi><mo>−</mo><mi mathvariant="normal">ℓ</mi></mrow><annotation encoding="application/x-tex">x - \ell</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord">ℓ</span></span></span></span></span> ce qui va peut-être sortir de l’espace. T’as une idée pour gérer ça ? Je sais pas si je suis très clair. <img src="/static/smileys/smile.png" alt=":)" class="smiley"></p>
<p>Je vais essayer ces différentes conditions aux bords. En modifiant vite fait, j’ai à peu près le même résultat qu’avant, mais il faut sûrement que je revoie mon implémentation en Python.</p>Résolution numérique d’une équation aux dérivées partielles, message #1994292019-02-23T15:39:06+01:00adri1/@adri1https://zestedesavoir.com/forums/sujet/12143/resolution-numerique-dune-equation-aux-derivees-partielles/?page=1#p199429<p>OK, ça m’a pas l’air absurde. Deux conseils :</p>
<ul>
<li>fais des différences finies centrées pour la dérivée spatiale <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="normal">∂</mi><mi>x</mi></msub><mi>h</mi><mo>≈</mo><mfrac><mrow><msub><mi>h</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>h</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><mrow><mn>2</mn><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\partial_xh\approx\dfrac{h_{i+1}-h_{i-1}}{2\Delta x}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.05556em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathdefault">h</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.05744em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord">Δ</span><span class="mord mathdefault">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>, tu gagnes un ordre de précision sans calculer plus et ça évite de biaiser ton estimation dans une direction ou l’autre ;</li>
<li>ne prend pas <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">ℓ</mi><mo>=</mo><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\ell=\Delta x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord">ℓ</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord">Δ</span><span class="mord mathdefault">x</span></span></span></span></span>, <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">ℓ</mi></mrow><annotation encoding="application/x-tex">\ell</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord">ℓ</span></span></span></span></span> est un paramètre physique en soit et tu n’as pas intérêt à l’avoir dépendant du pas que tu as choisis.</li>
</ul>
<p>Par ailleurs, Neumann comme condition au bord me parait pas forcément judicieux, ça t’empêche d’avoir un flux de sable. Tu pourrais imposer un niveau, ou alors éventuellement un flux non nul qui dépend du vent.</p>Résolution numérique d’une équation aux dérivées partielles, message #1994262019-02-23T14:57:47+01:00Tintinfil/@Tintinfilhttps://zestedesavoir.com/forums/sujet/12143/resolution-numerique-dune-equation-aux-derivees-partielles/?page=1#p199426<p>C’est sûrement pas bien fait, mais voici comment j’ai mis en place la méthode explicite. Un peu au hasard, j’ai pris les conditions de Neumann aux bords. J’ai d’abord discrétisé l’espace puis le temps. Comme je ne savais pas comment gérer le terme entre crochet, j’ai pris <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">ℓ</mi></mrow><annotation encoding="application/x-tex">\ell</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord">ℓ</span></span></span></span></span> (notée <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="normal">ℓ</mi><mtext>r</mtext></msub></mrow><annotation encoding="application/x-tex">\ell_\text r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">ℓ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">r</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span> dans mon texte) égale au pas de l’espace et j’ai bidouillé ensuite avec les conditions aux bords. Je te mets ci-dessous ce que j’ai fait.</p>
<div class="custom-block custom-block-spoiler"><div class="custom-block-body"><figure><img src="/media/galleries/464/8ba4b116-6d97-4038-8235-8eadba0b7d20.png" alt="Schéma"><figcaption>Schéma</figcaption></figure></div></div>
<p>J’ai déjà simulé la formation d’un tas de simple grâce aux éléments distincts.</p>Résolution numérique d’une équation aux dérivées partielles, message #1994222019-02-23T13:26:28+01:00adri1/@adri1https://zestedesavoir.com/forums/sujet/12143/resolution-numerique-dune-equation-aux-derivees-partielles/?page=1#p199422<blockquote>
<p>Euler explicite pour une EDP ?</p>
</blockquote>
<p>J’imagine qu’il parle de l’intégration en temps, et c’est pas forcément un problème si on fait gaffe à la stabilité.</p>
<p>FreeFEM est une galère à utiliser si tu ne connais pas un minimum les éléments finis (d’ailleurs bonjour la galère pour écrire la forme faible du problème), par contre implémenter un schéma aux différences finis est très simple.</p>
<p><a href="/membres/voir/Teguad/" rel="nofollow" class="ping ping-link">@<span class="ping-username">Teguad</span></a> : tu peux nous écrire ton schéma numérique et nous expliciter ce que tu as pris comme conditions aux limites ?</p>
<p>EDIT : l’autre option est de laisser tomber cette équation et de simuler un tas de sable avec des éléments discrets.</p>Résolution numérique d’une équation aux dérivées partielles, message #1994212019-02-23T13:22:26+01:00Tintinfil/@Tintinfilhttps://zestedesavoir.com/forums/sujet/12143/resolution-numerique-dune-equation-aux-derivees-partielles/?page=1#p199421<p>Oui Euler explicite pour des ÉDP, c’est pas l’idéal. <img src="/static/smileys/heureux.png" alt=":D" class="smiley"> Je vais essayer de me documenter sur ces méthodes pour voir si je peux en faire quelque chose. Sinon, je ne connaissais pas FreeFem, donc pourquoi pas. Merci pour ton aide !</p>Résolution numérique d’une équation aux dérivées partielles, message #1994152019-02-23T12:43:16+01:00anonyme/@anonymehttps://zestedesavoir.com/forums/sujet/12143/resolution-numerique-dune-equation-aux-derivees-partielles/?page=1#p199415<p>Euler explicite pour une EDP ?
Pour la résolution des EDP il y a 3 grandes méthodes :</p>
<ul>
<li>les différences finies,</li>
<li>les éléments finis,</li>
<li>les volumes finis</li>
</ul>
<p>Je ne suis pas sûr qu’il soit très accessible à ton niveau mais tu peux essayer de voir pour utiliser freefem pour résoudre ton équation. Le faire toi même sera encore plus compliqué.</p>Résolution numérique d’une équation aux dérivées partielles, message #1994092019-02-23T11:59:45+01:00Tintinfil/@Tintinfilhttps://zestedesavoir.com/forums/sujet/12143/resolution-numerique-dune-equation-aux-derivees-partielles/?page=1#p199409<p>Bonjour !</p>
<p>Dans le cadre de mon TIPE, je m’intéresse à la modélisation des phénomènes de transport dans une dune de sable. Après plusieurs recherches, je suis tombé sur une équation régissant les phénomènes de reptation et de saltation<sup id="fnref-1-DYTFTgi3E"><a href="#fn-1-DYTFTgi3E" class="footnote-ref">1</a></sup>. Cette équation est<sup id="fnref-2-DYTFTgi3E"><a href="#fn-2-DYTFTgi3E" class="footnote-ref">2</a></sup>
<span class="inlineMath inlineMathDouble"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="normal">∂</mi><mi>t</mi></msub><mi>h</mi><mo>+</mo><mi>A</mi><msubsup><mrow><mo fence="true">[</mo><mfrac><mrow><mi>B</mi><mo>+</mo><msub><mi mathvariant="normal">∂</mi><mi>x</mi></msub><mi>h</mi></mrow><msqrt><mrow><mn>1</mn><mo>+</mo><mo>(</mo><msub><mi mathvariant="normal">∂</mi><mi>x</mi></msub><mi>h</mi><msup><mo>)</mo><mn>2</mn></msup></mrow></msqrt></mfrac><mo fence="true">]</mo></mrow><mrow><mi>x</mi><mo>−</mo><mi mathvariant="normal">ℓ</mi></mrow><mi>x</mi></msubsup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\partial_t h + A \left[\frac{B + \partial_x h}{\sqrt{1 + (\partial_x h)^2}}\right]_{x - \ell}^x = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.05556em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathdefault">h</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:3.1623529999999995em;vertical-align:-1.358061em;"></span><span class="mord mathdefault">A</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">[</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em;"><span style="top:-2.175em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.935em;"><span class="svg-align" style="top:-3.2em;"><span class="pstrut" style="height:3.2em;"></span><span class="mord" style="padding-left:1em;"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.05556em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathdefault">h</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-2.8950000000000005em;"><span class="pstrut" style="height:3.2em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.28em;"><svg width="400em" height="1.28em" viewBox="0 0 400000 1296" preserveAspectRatio="xMinYMin slice"><path d="M263,681c0.7,0,18,39.7,52,119c34,79.3,68.167,
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où <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>h</mi><mspace></mspace><mspace width="0.1111111111111111em"></mspace><mtext></mtext><mo>:</mo><mspace width="0.3333333333333333em"></mspace><mo>(</mo><mi>x</mi><mo separator="true">,</mo><mi>t</mi><mo>)</mo><mo>∈</mo><mi mathvariant="double-struck">R</mi><mo>×</mo><mrow><mo fence="true">[</mo><mn>0</mn><mo separator="true">,</mo><mo>+</mo><mi mathvariant="normal">∞</mi><mo fence="true">[</mo></mrow><mo>⟼</mo><mi>h</mi><mo>(</mo><mi>x</mi><mo separator="true">,</mo><mi>t</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">h \colon (x, t) \in \mathbb R \times \left[0, +\infty\right[ \longmapsto h(x, t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">h</span><span class="mspace nobreak"></span><span class="mspace" style="margin-right:0.1111111111111111em;"></span><span class="mpunct"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mspace" style="margin-right:-0.16666666666666666em;"></span><span class="mord"><span class="mrel">:</span></span><span class="mspace" style="margin-right:0.3333333333333333em;"></span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.77222em;vertical-align:-0.08333em;"></span><span class="mord mathbb">R</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">+</span><span class="mord">∞</span><span class="mclose delimcenter" style="top:0em;">[</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⟼</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">h</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">t</span><span class="mclose">)</span></span></span></span></span> est la fonction solution et où <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault">A</span></span></span></span></span>, <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.05017em;">B</span></span></span></span></span> et <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">ℓ</mi></mrow><annotation encoding="application/x-tex">\ell</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord">ℓ</span></span></span></span></span> sont trois réels positifs. Pour remettre en contexte, cette équation s’applique dans un milieu granulaire (typiquement une dune de sable) où un vent arrive à un angle constant sur le sol et elle propose de modéliser les rides de sable observées. La quantité <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>h</mi><mo>(</mo><mi>x</mi><mo separator="true">,</mo><mi>t</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">h(x, t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">h</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">t</span><span class="mclose">)</span></span></span></span></span> est la hauteur de sable à la position <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span></span></span></span></span> et à l’instant <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61508em;vertical-align:0em;"></span><span class="mord mathdefault">t</span></span></span></span></span>.</p>
<p>Afin de vérifier si elle décrit bien le phénomène réel, je voudrais la simuler numériquement. J’ai essayé de mettre en place un schéma d’Euler explicite, mais le résultat n’est pas concluant. De plus, je ne sais pas comment gérer la différence entre crochet au niveau des bords. En effet, pour <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span>, il va falloir évaluer <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="normal">∂</mi><mi>x</mi></msub><mi>h</mi></mrow><annotation encoding="application/x-tex">\partial_x h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.05556em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathdefault">h</span></span></span></span></span> en <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi><mo>−</mo><mi mathvariant="normal">ℓ</mi><mo>=</mo><mo>−</mo><mi mathvariant="normal">ℓ</mi></mrow><annotation encoding="application/x-tex">x - \ell = -\ell</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord">ℓ</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em;"></span><span class="mord">−</span><span class="mord">ℓ</span></span></span></span></span> et cette quantité n’existera pas. J’ai contacté l’auteur et il n’a jamais simulé l’équation. De plus, aucune condition aux bords n’est précisée.</p>
<p>D’où mes questions. Avez-vous déjà fait face à une telle équation ? Et comment peut-on la simuler, <em>i. e.</em> comment peut-on gérer le terme entre crochet au niveau des bords ? Je précise que <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">ℓ</mi></mrow><annotation encoding="application/x-tex">\ell</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord">ℓ</span></span></span></span></span> est normalement assez petite (de l’ordre de quelques diamètres d’un grain).</p>
<p>Merci beaucoup ! <3</p>
<div class="footnotes">
<hr>
<ol>
<li id="fn-1-DYTFTgi3E">
<p>Cf. <a href="https://tel.archives-ouvertes.fr/tel-00199032">https://tel.archives-ouvertes.fr/tel-00199032</a>, page 143</p>
<a href="#fnref-1-DYTFTgi3E" class="footnote-backref" title="Retourner au texte de la note 1">↩</a>
</li>
<li id="fn-2-DYTFTgi3E">
<p>On a noté <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>[</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo separator="true">,</mo><mi>t</mi><mo>)</mo><msubsup><mo>]</mo><mrow><mi>x</mi><mo>−</mo><mi mathvariant="normal">ℓ</mi></mrow><mi>x</mi></msubsup><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo separator="true">,</mo><mi>t</mi><mo>)</mo><mo>−</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>−</mo><mi mathvariant="normal">ℓ</mi><mo separator="true">,</mo><mi>t</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">[f(x, t)]_{x - \ell}^x = f(x, t) - f(x - \ell, t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0914389999999998em;vertical-align:-0.34143899999999994em;"></span><span class="mopen">[</span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">t</span><span class="mclose">)</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-2.4168920000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">x</span><span class="mbin mtight">−</span><span class="mord mtight">ℓ</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.34143899999999994em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">ℓ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">t</span><span class="mclose">)</span></span></span></span></span>.</p>
<a href="#fnref-2-DYTFTgi3E" class="footnote-backref" title="Retourner au texte de la note 2">↩</a>
</li>
</ol>
</div>Linéarisation d’une équation aux dérivées partielles, message #1768222018-04-02T13:24:35+02:00Tintinfil/@Tintinfilhttps://zestedesavoir.com/forums/sujet/10385/linearisation-dune-equation-aux-derivees-partielles/?page=1#p176822<blockquote>
<p>J’avoue avoir beaucoup de mal à comprendre pourquoi cette formulation te convient plus que celle qui consiste à dire avec des mots qu’on linéarise et qu’on regarde le signe du taux de croissance parce qu’elles sont strictement équivalente, il n’y en a pas une plus rigoureuse ou générale que l’autre, ce que tu as écrit est exactement la même chose et les conditions d’applicabilité sont les mêmes.</p>
</blockquote>
<p>C’est ce que mon prof me demande. À la base, le chercheur qui faisait sa conférence l’a fait comme tu me le proposais (pour coller à notre niveau L1 en maths) et son argumentation me convenait, mais mon prof n’était pas convaincu par ce type d’arguments et ce même chercheur n’a pas montré ce résultat comme cela lors de ses recherches. C’est tout ! Crois-moi, je ne voulais pas mettre en doute tes affirmations. Mais, les énoncés de ce livre ne sont pas triviaux, ce qui me fait penser que mon problème n’était pas si simple qu’il n’en avait l’air.</p>Linéarisation d’une équation aux dérivées partielles, message #1768202018-04-02T13:10:06+02:00adri1/@adri1https://zestedesavoir.com/forums/sujet/10385/linearisation-dune-equation-aux-derivees-partielles/?page=1#p176820<p>J’avoue avoir beaucoup de mal à comprendre pourquoi cette formulation te convient plus que celle qui consiste à dire avec des mots qu’on linéarise et qu’on regarde le signe du taux de croissance parce qu’elles sont strictement équivalente, il n’y en a pas une plus rigoureuse ou générale que l’autre, ce que tu as écrit est exactement la même chose et les conditions d’applicabilité sont les mêmes.</p>
<p>Si dans le bouquin il n’y a pas mention de Taylor ou de notations de Landau, c’est parce que ce sont des choses triviales que tout le monde comprend lorsqu’on parle de faire une analyse de stabilité linéaire. Si tu reprends mon tout premier commentaire, tu verras que je n’en fais d’ailleurs pas mention, c’est toi qui a absolument voulu raccrocher les wagons avec ces notions (ce qui est parfaitement légitime, mais clairement pas nécessaire).</p>