SherbyNet - /page/wikibook/entry.mu

Micron Document

<!DOCTYPE html>

<title>Integrable algorithm</title>

</head>

<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Integrable algorithm</span></span>

</h1>

</header>

/* start https://en.wikipedia.org/ */

.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}

/* end https://en.wikipedia.org/ */

</style>

<p><b>Integrable algorithms</b> are numerical algorithms that rely on basic ideas from the mathematical theory of <a href="Integrable_systems" class="mw-redirect" title="Integrable systems">integrable systems</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Background">Background</h2></div>

<p>The theory of integrable systems has advanced with the connection between <a href="Numerical_analysis" title="Numerical analysis">numerical analysis</a>. For example, the discovery of solitons came from the numerical experiments to the <a href="KdV_equation" class="mw-redirect" title="KdV equation">KdV equation</a> by <a href="Norman_Zabusky" title="Norman Zabusky">Norman Zabusky</a> and <a href="Martin_David_Kruskal" title="Martin David Kruskal">Martin David Kruskal</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Today, various relations between numerical analysis and integrable systems have been found (<a href="Toda_lattice" title="Toda lattice">Toda lattice</a> and <a href="Numerical_linear_algebra" title="Numerical linear algebra">numerical linear algebra</a>,<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> discrete soliton equations and <a href="Series_acceleration" title="Series acceleration">series acceleration</a><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>), and studies to apply integrable systems to numerical computation are rapidly advancing.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Integrable_difference_schemes">Integrable difference schemes</h2></div>

<p>Generally, it is hard to accurately compute the solutions of nonlinear differential equations due to its non-linearity. In order to overcome this difficulty, R. Hirota has made discrete versions of integrable systems with the viewpoint of "Preserve mathematical structures of integrable systems in the discrete versions".<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup>

</p><p>At the same time, <a href="Mark_J._Ablowitz" title="Mark J. Ablowitz">Mark J. Ablowitz</a> and others have not only made discrete soliton equations with discrete <a href="Lax_pair" title="Lax pair">Lax pair</a> but also compared numerical results between integrable difference schemes and ordinary methods.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> As a result of their experiments, they have found that the accuracy can be improved with integrable difference schemes at some cases.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}

/* end https://en.wikipedia.org/ */

</style><div class="reflist">

/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFNakamura2004" class="citation conference cs1">Nakamura, Y. (2004). <i>A new approach to numerical algorithms in terms of integrable systems</i>. International Conference on Informatics Research for Development of Knowledge Society Infrastructure. IEEE. pp. <span class="nowrap">194–</span>205. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2Ficks.2004.1313425">10.1109/icks.2004.1313425</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-7695-2150-9</bdi>.</cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFZabuskyKruskal1965" class="citation journal cs1">Zabusky, N. J.; Kruskal, M. D. (1965-08-09). <a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Fphysrevlett.15.240">"Interaction of "Solitons" in a Collisionless Plasma and the Recurrence of Initial States"</a>. <i>Physical Review Letters</i>. <b>15</b> (6). American Physical Society (APS): <span class="nowrap">240–</span>243. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1965PhRvL..15..240Z">1965PhRvL..15..240Z</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Fphysrevlett.15.240">10.1103/physrevlett.15.240</a></span>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0031-9007">0031-9007</a>.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFSogo1993" class="citation journal cs1">Sogo, Kiyoshi (1993-04-15). "Toda Molecule Equation and Quotient-Difference Method". <i>Journal of the Physical Society of Japan</i>. <b>62</b> (4). Physical Society of Japan: <span class="nowrap">1081–</span>1084. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1993JPSJ...62.1081S">1993JPSJ...62.1081S</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1143%2Fjpsj.62.1081">10.1143/jpsj.62.1081</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0031-9015">0031-9015</a>.</cite></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFIwasakiNakamura2006" class="citation journal cs1">Iwasaki, Masashi; Nakamura, Yoshimasa (2006). <a rel="nofollow" class="external text" href="http://projecteuclid.org/euclid.jjiam/1197390799">"Accurate computation of singular values in terms of shifted integrable schemes"</a>. <i>Japan Journal of Industrial and Applied Mathematics</i>. <b>23</b> (3). Springer Science and Business Media LLC: <span class="nowrap">239–</span>259. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf03167593">10.1007/bf03167593</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0916-7005">0916-7005</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121824363">121824363</a>.</cite></span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFPapageorgiouGrammaticosRamani1993" class="citation journal cs1">Papageorgiou, V.; Grammaticos, B.; Ramani, A. (1993). "Integrable lattices and convergence acceleration algorithms". <i>Physics Letters A</i>. <b>179</b> (2). Elsevier BV: <span class="nowrap">111–</span>115. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1993PhLA..179..111P">1993PhLA..179..111P</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0375-9601%2893%2990658-m">10.1016/0375-9601(93)90658-m</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0375-9601">0375-9601</a>.</cite></span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite id="CITEREFChangHeHuLi2017" class="citation journal cs1">Chang, Xiang-Ke; He, Yi; Hu, Xing-Biao; Li, Shi-Hao (2017-07-01). "A new integrable convergence acceleration algorithm for computing Brezinski–Durbin–Redivo-Zaglia's sequence transformation via pfaffians". <i>Numerical Algorithms</i>. <b>78</b> (1). Springer Science and Business Media LLC: <span class="nowrap">87–</span>106. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs11075-017-0368-z">10.1007/s11075-017-0368-z</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1017-1398">1017-1398</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:4974630">4974630</a>.</cite></span>

</li>

<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFNakamura2001" class="citation journal cs1">Nakamura, Yoshimasa (2001). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fs0377-0427%2800%2900316-2">"Algorithms associated with arithmetic, geometric and harmonic means and integrable systems"</a>. <i>Journal of Computational and Applied Mathematics</i>. <b>131</b> (<span class="nowrap">1–</span>2). Elsevier BV: <span class="nowrap">161–</span>174. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2001JCoAM.131..161N">2001JCoAM.131..161N</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fs0377-0427%2800%2900316-2">10.1016/s0377-0427(00)00316-2</a></span>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0377-0427">0377-0427</a>.</cite></span>

</li>

<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><cite id="CITEREFChu2008" class="citation journal cs1">Chu, Moody T. (2008-04-25). "Linear algebra algorithms as dynamical systems". <i>Acta Numerica</i>. <b>17</b>. Cambridge University Press (CUP): <span class="nowrap">1–</span>86. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2Fs0962492906340019">10.1017/s0962492906340019</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0962-4929">0962-4929</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:8746366">8746366</a>.</cite></span>

</li>

<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><cite id="CITEREFHirota1977" class="citation journal cs1">Hirota, Ryogo (1977-10-15). "Nonlinear Partial Difference Equations. I. A Difference Analogue of the Korteweg-de Vries Equation". <i>Journal of the Physical Society of Japan</i>. <b>43</b> (4). Physical Society of Japan: <span class="nowrap">1424–</span>1433. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1977JPSJ...43.1424H">1977JPSJ...43.1424H</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1143%2Fjpsj.43.1424">10.1143/jpsj.43.1424</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0031-9015">0031-9015</a>.</cite></span>

</li>

<li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><cite id="CITEREFHirota1977" class="citation journal cs1">Hirota, Ryogo (1977-12-15). "Nonlinear Partial Difference Equations. II. Discrete-Time Toda Equation". <i>Journal of the Physical Society of Japan</i>. <b>43</b> (6). Physical Society of Japan: <span class="nowrap">2074–</span>2078. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1977JPSJ...43.2074H">1977JPSJ...43.2074H</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1143%2Fjpsj.43.2074">10.1143/jpsj.43.2074</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0031-9015">0031-9015</a>.</cite></span>

</li>

<li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><cite id="CITEREFHirota1977" class="citation journal cs1">Hirota, Ryogo (1977-12-15). "Nonlinear Partial Difference Equations III; Discrete Sine-Gordon Equation". <i>Journal of the Physical Society of Japan</i>. <b>43</b> (6). Physical Society of Japan: <span class="nowrap">2079–</span>2086. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1977JPSJ...43.2079H">1977JPSJ...43.2079H</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1143%2Fjpsj.43.2079">10.1143/jpsj.43.2079</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0031-9015">0031-9015</a>.</cite></span>

</li>

<li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><cite id="CITEREFHirota1978" class="citation journal cs1">Hirota, Ryogo (1978-07-15). "Nonlinear Partial Difference Equations. IV. Bäcklund Transformation for the Discrete-Time Toda Equation". <i>Journal of the Physical Society of Japan</i>. <b>45</b> (1). Physical Society of Japan: <span class="nowrap">321–</span>332. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1978JPSJ...45..321H">1978JPSJ...45..321H</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1143%2Fjpsj.45.321">10.1143/jpsj.45.321</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0031-9015">0031-9015</a>.</cite></span>

</li>

<li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><cite id="CITEREFHirota1979" class="citation journal cs1">Hirota, Ryogo (1979-01-15). "Nonlinear Partial Difference Equations. V. Nonlinear Equations Reducible to Linear Equations". <i>Journal of the Physical Society of Japan</i>. <b>46</b> (1). Physical Society of Japan: <span class="nowrap">312–</span>319. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1979JPSJ...46..312H">1979JPSJ...46..312H</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1143%2Fjpsj.46.312">10.1143/jpsj.46.312</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0031-9015">0031-9015</a>.</cite></span>

</li>

<li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><cite id="CITEREFAblowitzLadik1975" class="citation journal cs1">Ablowitz, M. J.; Ladik, J. F. (1975). "Nonlinear differential−difference equations". <i>Journal of Mathematical Physics</i>. <b>16</b> (3). AIP Publishing: <span class="nowrap">598–</span>603. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1975JMP....16..598A">1975JMP....16..598A</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.522558">10.1063/1.522558</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0022-2488">0022-2488</a>.</cite></span>

</li>

<li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><cite id="CITEREFAblowitzLadik1976" class="citation journal cs1">Ablowitz, M. J.; Ladik, J. F. (1976). "Nonlinear differential–difference equations and Fourier analysis". <i>Journal of Mathematical Physics</i>. <b>17</b> (6). AIP Publishing: <span class="nowrap">1011–</span>1018. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1976JMP....17.1011A">1976JMP....17.1011A</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.523009">10.1063/1.523009</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0022-2488">0022-2488</a>.</cite></span>

</li>

<li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><cite id="CITEREFAblowitzLadik1976" class="citation journal cs1">Ablowitz, M. J.; Ladik, J. F. (1976). "A Nonlinear Difference Scheme and Inverse Scattering". <i>Studies in Applied Mathematics</i>. <b>55</b> (3). Wiley: <span class="nowrap">213–</span>229. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fsapm1976553213">10.1002/sapm1976553213</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0022-2526">0022-2526</a>.</cite></span>

</li>

<li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><cite id="CITEREFAblowitzLadik1977" class="citation journal cs1">Ablowitz, M. J.; Ladik, J. F. (1977). "On the Solution of a Class of Nonlinear Partial Difference Equations". <i>Studies in Applied Mathematics</i>. <b>57</b> (1). Wiley: <span class="nowrap">1–</span>12. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fsapm19775711">10.1002/sapm19775711</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0022-2526">0022-2526</a>.</cite></span>

</li>

<li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><cite id="CITEREFAblowitzSegur1981" class="citation book cs1">Ablowitz, Mark J.; Segur, Harvey (1981). <i>Solitons and the Inverse Scattering Transform</i>. Philadelphia: <a href="Society_for_Industrial_and_Applied_Mathematics" title="Society for Industrial and Applied Mathematics">Society for Industrial and Applied Mathematics</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F1.9781611970883">10.1137/1.9781611970883</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-89871-174-5</bdi>.</cite></span>

</li>

<li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><cite id="CITEREFTahaAblowitz1984" class="citation journal cs1">Taha, Thiab R; Ablowitz, Mark J (1984). "Analytical and numerical aspects of certain nonlinear evolution equations. I. Analytical". <i>Journal of Computational Physics</i>. <b>55</b> (2). Elsevier BV: <span class="nowrap">192–</span>202. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1984JCoPh..55..192T">1984JCoPh..55..192T</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0021-9991%2884%2990002-0">10.1016/0021-9991(84)90002-0</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0021-9991">0021-9991</a>.</cite></span>

</li>

<li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><cite id="CITEREFTahaAblowitz1984" class="citation journal cs1">Taha, Thiab R; Ablowitz, Mark I (1984). "Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation". <i>Journal of Computational Physics</i>. <b>55</b> (2). Elsevier BV: <span class="nowrap">203–</span>230. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1984JCoPh..55..203T">1984JCoPh..55..203T</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0021-9991%2884%2990003-2">10.1016/0021-9991(84)90003-2</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0021-9991">0021-9991</a>.</cite></span>

</li>

<li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><cite id="CITEREFTahaAblowitz1984" class="citation journal cs1">Taha, Thiab R; Ablowitz, Mark I (1984). "Analytical and numerical aspects of certain nonlinear evolution equations. III. Numerical, Korteweg-de Vries equation". <i>Journal of Computational Physics</i>. <b>55</b> (2). Elsevier BV: <span class="nowrap">231–</span>253. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1984JCoPh..55..231T">1984JCoPh..55..231T</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0021-9991%2884%2990004-4">10.1016/0021-9991(84)90004-4</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0021-9991">0021-9991</a>.</cite></span>

</li>

<li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><cite id="CITEREFTahaAblowitz1988" class="citation journal cs1">Taha, Thiab R; Ablowitz, Mark J (1988). "Analytical and numerical aspects of certain nonlinear evolution equations IV. Numerical, modified Korteweg-de Vries equation". <i>Journal of Computational Physics</i>. <b>77</b> (2). Elsevier BV: <span class="nowrap">540–</span>548. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1988JCoPh..77..540T">1988JCoPh..77..540T</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0021-9991%2888%2990184-2">10.1016/0021-9991(88)90184-2</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0021-9991">0021-9991</a>.</cite></span>

</li>

</ol></div></div>

<ul><li><a href="Soliton" title="Soliton">Soliton</a></li>

<li><a href="Integrable_system" title="Integrable system">Integrable system</a></li></ul>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}

/* end https://en.wikipedia.org/ */

</style><style data-mw-deduplicate="TemplateStyles:r1236075235">

/* start https://en.wikipedia.org/ */

.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}

/* end https://en.wikipedia.org/ */

</style></div><div role="navigation" class="navbox" aria-labelledby="Numerical_methods_for_partial_differential_equations284" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><style data-mw-deduplicate="TemplateStyles:r1239400231">

/* start https://en.wikipedia.org/ */

.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}

/* end https://en.wikipedia.org/ */

</style><div id="Numerical_methods_for_partial_differential_equations284" style="font-size:114%;margin:0 4em"><a href="Numerical_methods_for_partial_differential_equations" title="Numerical methods for partial differential equations">Numerical methods for partial differential equations</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Finite_difference_method" title="Finite difference method">Finite difference</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Parabolic_partial_differential_equation" title="Parabolic partial differential equation">Parabolic</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="FTCS_scheme" title="FTCS scheme">Forward-time central-space</a> (FTCS)</li>

<li><a href="Crank%E2%80%93Nicolson_method" title="Crank–Nicolson method">Crank–Nicolson</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Hyperbolic_partial_differential_equation" title="Hyperbolic partial differential equation">Hyperbolic</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Lax%E2%80%93Friedrichs_method" title="Lax–Friedrichs method">Lax–Friedrichs</a></li>

<li><a href="Lax%E2%80%93Wendroff_method" title="Lax–Wendroff method">Lax–Wendroff</a></li>

<li><a href="MacCormack_method" title="MacCormack method">MacCormack</a></li>

<li><a href="Upwind_scheme" title="Upwind scheme">Upwind</a></li>

<li><a href="Method_of_characteristics" title="Method of characteristics">Method of characteristics</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Others</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Alternating_direction_implicit_method" class="mw-redirect" title="Alternating direction implicit method">Alternating direction-implicit</a> (ADI)</li>

<li><a href="Finite-difference_frequency-domain_method" title="Finite-difference frequency-domain method">Finite-difference frequency-domain</a> (FDFD)</li>

<li><a href="Finite-difference_time-domain_method" title="Finite-difference time-domain method">Finite-difference time-domain</a> (FDTD)</li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Finite_volume_method" title="Finite volume method">Finite volume</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Godunov's_scheme" title="Godunov's scheme">Godunov</a></li>

<li><a href="High-resolution_scheme" title="High-resolution scheme">High-resolution</a></li>

<li><a href="MUSCL_scheme" title="MUSCL scheme">Monotonic upstream-centered</a> (MUSCL)</li>

<li><a href="AUSM" class="mw-redirect" title="AUSM">Advection upstream-splitting</a> (AUSM)</li>

<li><a href="Riemann_solver" title="Riemann solver">Riemann solver</a></li>

<li><a href="ENO_methods" title="ENO methods">Essentially non-oscillatory</a> (ENO)</li>

<li><a href="WENO_methods" title="WENO methods">Weighted essentially non-oscillatory</a> (WENO)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Finite_element_method" title="Finite element method">Finite element</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Extended_finite_element_method" title="Extended finite element method">Extended</a> (XFEM)</li>

<li><a href="Discontinuous_Galerkin_method" title="Discontinuous Galerkin method">Discontinuous Galerkin</a> (DG)</li>

<li><a href="Spectral_element_method" title="Spectral element method">Spectral element</a> (SEM)</li>

<li><a href="Mortar_methods" title="Mortar methods">Mortar</a></li>

<li><a href="Gradient_discretisation_method" title="Gradient discretisation method">Gradient discretisation</a> (GDM)</li>

<li><a href="Loubignac_iteration" title="Loubignac iteration">Loubignac iteration</a></li>

<li><a href="Smoothed_finite_element_method" title="Smoothed finite element method">Smoothed</a> (S-FEM)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Meshfree_methods" title="Meshfree methods">Meshless/Meshfree</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Smoothed-particle_hydrodynamics" title="Smoothed-particle hydrodynamics">Smoothed-particle hydrodynamics</a> (SPH)</li>

<li><a href="Peridynamics" title="Peridynamics">Peridynamics</a> (PD)</li>

<li><a href="Moving_particle_semi-implicit_method" title="Moving particle semi-implicit method">Moving particle semi-implicit method</a> (MPS)</li>

<li><a href="Material_point_method" title="Material point method">Material point method</a> (MPM)</li>

<li><a href="Particle-in-cell" title="Particle-in-cell">Particle-in-cell</a> (PIC)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Domain_decomposition_methods" title="Domain decomposition methods">Domain decomposition</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Schur_complement_method" title="Schur complement method">Schur complement</a></li>

<li><a href="Fictitious_domain_method" title="Fictitious domain method">Fictitious domain</a></li>

<li><a href="Schwarz_alternating_method" title="Schwarz alternating method">Schwarz alternating</a>

<ul><li><a href="Additive_Schwarz_method" title="Additive Schwarz method">additive</a></li>

<li><a href="Abstract_additive_Schwarz_method" title="Abstract additive Schwarz method">abstract additive</a></li></ul></li>

<li><a href="Neumann%E2%80%93Dirichlet_method" title="Neumann–Dirichlet method">Neumann–Dirichlet</a></li>

<li><a href="Neumann%E2%80%93Neumann_methods" title="Neumann–Neumann methods">Neumann–Neumann</a></li>

<li><a href="Poincar%C3%A9%E2%80%93Steklov_operator" title="Poincaré–Steklov operator">Poincaré–Steklov operator</a></li>

<li><a href="Balancing_domain_decomposition_method" title="Balancing domain decomposition method">Balancing</a> (BDD)</li>

<li><a href="BDDC" title="BDDC">Balancing by constraints</a> (BDDC)</li>

<li><a href="FETI" title="FETI">Tearing and interconnect</a> (FETI)</li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Others</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Spectral_method" title="Spectral method">Spectral</a></li>

<li><a href="Pseudo-spectral_method" title="Pseudo-spectral method">Pseudospectral</a> (DVR)</li>

<li><a href="Method_of_lines" title="Method of lines">Method of lines</a></li>

<li><a href="Multigrid_method" title="Multigrid method">Multigrid</a></li>

<li><a href="Collocation_method" title="Collocation method">Collocation</a></li>

<li><a href="Level-set_method" title="Level-set method">Level-set</a></li>

<li><a href="Boundary_element_method" title="Boundary element method">Boundary element</a>

<ul><li><a href="Method_of_moments_(electromagnetics)" title="Method of moments (electromagnetics)">Method of moments</a></li></ul></li>

<li><a href="Immersed_boundary_method" title="Immersed boundary method">Immersed boundary</a></li>

<li><a href="Analytic_element_method" title="Analytic element method">Analytic element</a></li>

<li><a href="Isogeometric_analysis" title="Isogeometric analysis">Isogeometric analysis</a></li>

<li><a href="Infinite_difference_method" title="Infinite difference method">Infinite difference method</a></li>

<li><a href="Infinite_element_method" title="Infinite element method">Infinite element method</a></li>

<li><a href="Galerkin_method" title="Galerkin method">Galerkin method</a>

<ul><li><a href="Petrov%E2%80%93Galerkin_method" title="Petrov–Galerkin method">Petrov–Galerkin method</a></li></ul></li>

<li><a href="Validated_numerics" title="Validated numerics">Validated numerics</a></li>

<li><a href="Computer-assisted_proof" title="Computer-assisted proof">Computer-assisted proof</a></li>

<li><a href="Method_of_fundamental_solutions" title="Method of fundamental solutions">Method of fundamental solutions</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Numerical_methods_for_ordinary_differential_equations" title="Numerical methods for ordinary differential equations">Numerical methods for ordinary differential equations</a></li>

<li><a href="Numerical_integration" title="Numerical integration">Numerical integration</a></li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2023-12-21" href="https://en.wikipedia.org/wiki/?title=Integrable_algorithm&oldid=1191087478">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>