Spectral method - Wikipedia

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searchaux" style="display:none">Class of methods used in numerical analysis and scientific computing to solve ODE/PDE</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.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}}</style><table class="box-More_footnotes_needed plainlinks metadata ambox ambox-style ambox-More_footnotes_needed" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article includes a list of <a href="/wiki/Wikipedia:Citing_sources#General_references" title="Wikipedia:Citing sources">general references</a>, but <b>it lacks sufficient corresponding <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a></b>.<span class="hide-when-compact"> Please help to <a href="/wiki/Wikipedia:WikiProject_Reliability" title="Wikipedia:WikiProject Reliability">improve</a> this article by <a href="/wiki/Wikipedia:When_to_cite" title="Wikipedia:When to cite">introducing</a> more precise citations.</span> <span class="date-container"><i>(<span class="date">August 2013</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p><b>Spectral methods</b> are a class of techniques used in <a href="/wiki/Applied_mathematics" title="Applied mathematics">applied mathematics</a> and <a href="/wiki/Scientific_computing" class="mw-redirect" title="Scientific computing">scientific computing</a> to numerically solve certain <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a>. The idea is to write the solution of the differential equation as a sum of certain "<a href="/wiki/Basis_function" title="Basis function">basis functions</a>" (for example, as a <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a> which is a sum of <a href="/wiki/Sine_wave" title="Sine wave">sinusoids</a>) and then to choose the coefficients in the sum in order to satisfy the differential equation as well as possible. </p><p>Spectral methods and <a href="/wiki/Finite-element_method" class="mw-redirect" title="Finite-element method">finite-element methods</a> are closely related and built on the same ideas; the main difference between them is that spectral methods use basis functions that are generally nonzero over the whole domain, while finite element methods use basis functions that are nonzero only on small subdomains (<a href="/wiki/Compact_support" class="mw-redirect" title="Compact support">compact support</a>). Consequently, spectral methods connect variables <i>globally</i> while finite elements do so <i>locally</i>. Partially for this reason, spectral methods have excellent error properties, with the so-called "exponential convergence" being the fastest possible, when the solution is <a href="/wiki/Smooth_function" class="mw-redirect" title="Smooth function">smooth</a>. However, there are no known three-dimensional single-domain spectral <a href="/wiki/Shock_capturing" class="mw-redirect" title="Shock capturing">shock capturing</a> results (shock waves are not smooth).<sup id="cite_ref-CHQZ_1-0" class="reference"><a href="#cite_note-CHQZ-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> In the finite-element community, a method where the degree of the elements is very high or increases as the grid parameter <i>h</i> increases is sometimes called a <a href="/wiki/Spectral-element_method" class="mw-redirect" title="Spectral-element method">spectral-element method</a>. </p><p>Spectral methods can be used to solve <a href="/wiki/Differential_equations" class="mw-redirect" title="Differential equations">differential equations</a> (PDEs, ODEs, eigenvalue, etc) and <a href="/wiki/Optimization_problem" title="Optimization problem">optimization problems</a>. When applying spectral methods to time-dependent PDEs, the solution is typically written as a sum of basis functions with time-dependent coefficients; substituting this in the PDE yields a system of ODEs in the coefficients which can be solved using any <a href="/wiki/Numerical_methods_for_ordinary_differential_equations" title="Numerical methods for ordinary differential equations">numerical method for ODEs</a>. Eigenvalue problems for ODEs are similarly converted to matrix eigenvalue problems <sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (August 2013)">citation needed</span></a></i>]</sup>. </p><p>Spectral methods were developed in a long series of papers by <a href="/wiki/Steven_Orszag" title="Steven Orszag">Steven Orszag</a> starting in 1969 including, but not limited to, Fourier series methods for periodic geometry problems, polynomial spectral methods for finite and unbounded geometry problems, pseudospectral methods for highly nonlinear problems, and spectral iteration methods for fast solution of steady-state problems. The implementation of the spectral method is normally accomplished either with <a href="/wiki/Collocation_method" title="Collocation method">collocation</a> or a <a href="/wiki/Galerkin_method" title="Galerkin method">Galerkin</a> or a <a href="/w/index.php?title=Tau_method&action=edit&redlink=1" class="new" title="Tau method (page does not exist)">Tau</a> approach . For very small problems, the spectral method is unique in that solutions may be written out symbolically, yielding a practical alternative to series solutions for differential equations. </p><p>Spectral methods can be computationally less expensive and easier to implement than finite element methods; they shine best when high accuracy is sought in simple domains with smooth solutions. However, because of their global nature, the matrices associated with step computation are dense and computational efficiency will quickly suffer when there are many degrees of freedom (with some exceptions, for example if matrix applications can be written as <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transforms</a>). For larger problems and nonsmooth solutions, finite elements will generally work better due to sparse matrices and better modelling of discontinuities and sharp bends. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Examples_of_spectral_methods">Examples of spectral methods</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spectral_method&action=edit&section=1" title="Edit section: Examples of spectral methods"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="A_concrete,_linear_example"><span id="A_concrete.2C_linear_example"></span>A concrete, linear example</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spectral_method&action=edit&section=2" title="Edit section: A concrete, linear example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Here we presume an understanding of basic multivariate <a href="/wiki/Calculus" title="Calculus">calculus</a> and <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a>. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf358a54b0375e22ae5f3ab2c3e1a22c0c87e11c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.444ex; height:2.843ex;" alt="{\displaystyle g(x,y)}"></span> is a known, complex-valued function of two real variables, and g is periodic in x and y (that is, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x,y)=g(x+2\pi ,y)=g(x,y+2\pi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x,y)=g(x+2\pi ,y)=g(x,y+2\pi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87181343594f4f4fa4b9731daae264b40731283e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.2ex; height:2.843ex;" alt="{\displaystyle g(x,y)=g(x+2\pi ,y)=g(x,y+2\pi )}"></span>) then we are interested in finding a function <i>f</i>(<i>x</i>,<i>y</i>) so that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}\right)f(x,y)=g(x,y)\quad {\text{for all }}x,y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for all </mtext> </mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}\right)f(x,y)=g(x,y)\quad {\text{for all }}x,y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96f7a0a7bc30e0fa16529d9664d4acaa783ce66b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:43.951ex; height:6.343ex;" alt="{\displaystyle \left({\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}\right)f(x,y)=g(x,y)\quad {\text{for all }}x,y}"></span></dd></dl> <p>where the expression on the left denotes the second partial derivatives of <i>f</i> in <i>x</i> and <i>y</i>, respectively. This is the <a href="/wiki/Poisson_equation" class="mw-redirect" title="Poisson equation">Poisson equation</a>, and can be physically interpreted as some sort of heat conduction problem, or a problem in potential theory, among other possibilities. </p><p>If we write <i>f</i> and <i>g</i> in Fourier series: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}f&=:\sum a_{j,k}e^{i(jx+ky)},\\[5mu]g&=:\sum b_{j,k}e^{i(jx+ky)},\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.578em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>f</mi> </mtd> <mtd> <mi></mi> <mo>=:</mo> <mo>∑</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">(</mo> <mi>j</mi> <mi>x</mi> <mo>+</mo> <mi>k</mi> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>g</mi> </mtd> <mtd> <mi></mi> <mo>=:</mo> <mo>∑</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">(</mo> <mi>j</mi> <mi>x</mi> <mo>+</mo> <mi>k</mi> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f&=:\sum a_{j,k}e^{i(jx+ky)},\\[5mu]g&=:\sum b_{j,k}e^{i(jx+ky)},\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b444d1620666b214445e57dcacc0c855753f3c0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:21.35ex; height:8.509ex;" alt="{\displaystyle {\begin{aligned}f&=:\sum a_{j,k}e^{i(jx+ky)},\\[5mu]g&=:\sum b_{j,k}e^{i(jx+ky)},\end{aligned}}}"></span></dd></dl> <p>and substitute into the differential equation, we obtain this equation: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum -a_{j,k}(j^{2}+k^{2})e^{i(jx+ky)}=\sum b_{j,k}e^{i(jx+ky)}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∑</mo> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">(</mo> <mi>j</mi> <mi>x</mi> <mo>+</mo> <mi>k</mi> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <mo>∑</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">(</mo> <mi>j</mi> <mi>x</mi> <mo>+</mo> <mi>k</mi> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum -a_{j,k}(j^{2}+k^{2})e^{i(jx+ky)}=\sum b_{j,k}e^{i(jx+ky)}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a82d3c53db7cab04368e693224d36493e6116990" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:44.104ex; height:3.843ex;" alt="{\displaystyle \sum -a_{j,k}(j^{2}+k^{2})e^{i(jx+ky)}=\sum b_{j,k}e^{i(jx+ky)}.}"></span></dd></dl> <p>We have exchanged partial differentiation with an infinite sum, which is legitimate if we assume for instance that <i>f</i> has a continuous second derivative. By the uniqueness theorem for Fourier expansions, we must then equate the Fourier coefficients term by term, giving </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{j,k}=-{\frac {b_{j,k}}{j^{2}+k^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mrow> <msup> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{j,k}=-{\frac {b_{j,k}}{j^{2}+k^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cec8da445516854ca0553cce50b968db8ac60384" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.314ex; height:6.343ex;" alt="{\displaystyle a_{j,k}=-{\frac {b_{j,k}}{j^{2}+k^{2}}}}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_*" class="reference nourlexpansion" style="font-weight:bold;">*</span>)</b></td></tr></tbody></table> <p>which is an explicit formula for the Fourier coefficients <i>a</i><sub><i>j</i>,<i>k</i></sub>. </p><p>With periodic boundary conditions, the <a href="/wiki/Poisson_equation" class="mw-redirect" title="Poisson equation">Poisson equation</a> possesses a solution only if <i>b</i><sub>0,0</sub> = 0. Therefore, we can freely choose <i>a</i><sub>0,0</sub> which will be equal to the mean of the resolution. This corresponds to choosing the integration constant. </p><p>To turn this into an algorithm, only finitely many frequencies are solved for. This introduces an error which can be shown to be proportional to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d6bbe9f9dd6a8e8aee377e3757cd0183bb31f18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.557ex; height:2.343ex;" alt="{\displaystyle h^{n}}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h:=1/n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>:=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h:=1/n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98ef1d1838c6b6f9b9d45980c4b9db0388abae59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.804ex; height:2.843ex;" alt="{\displaystyle h:=1/n}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> is the highest frequency treated. </p> <div class="mw-heading mw-heading4"><h4 id="Algorithm">Algorithm</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spectral_method&action=edit&section=3" title="Edit section: Algorithm"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol><li>Compute the Fourier transform (<i>b<sub>j,k</sub></i>) of <i>g</i>.</li> <li>Compute the Fourier transform (<i>a<sub>j,k</sub></i>) of <i>f</i> via the formula (<b><a href="#math_*">*</a></b>).</li> <li>Compute <i>f</i> by taking an inverse Fourier transform of (<i>a<sub>j,k</sub></i>).</li></ol> <p>Since we're only interested in a finite window of frequencies (of size <i>n</i>, say) this can be done using a <a href="/wiki/Fast_Fourier_transform" title="Fast Fourier transform">fast Fourier transform</a> algorithm. Therefore, globally the algorithm runs in <span class="nowrap">time <i>O</i>(<i>n</i> log <i>n</i>).</span> </p> <div class="mw-heading mw-heading3"><h3 id="Nonlinear_example">Nonlinear example</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spectral_method&action=edit&section=4" title="Edit section: Nonlinear example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>We wish to solve the forced, transient, nonlinear <a href="/wiki/Burgers%27_equation" title="Burgers' equation">Burgers' equation</a> using a spectral approach. </p><p>Given <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(x,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u(x,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/063b881cce5a211a628dd78879d4c87ecda374c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.665ex; height:2.843ex;" alt="{\displaystyle u(x,0)}"></span> on the periodic domain <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \left[0,2\pi \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mrow> <mo>[</mo> <mrow> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mi>π</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in \left[0,2\pi \right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8763ea878ea5cf7a7194c41dad8a6718150eda7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.413ex; height:2.843ex;" alt="{\displaystyle x\in \left[0,2\pi \right)}"></span>, find <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\in {\mathcal {U}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">U</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u\in {\mathcal {U}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aafb482628cdfbea0ecf0f757e7b6908c332e179" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.767ex; height:2.176ex;" alt="{\displaystyle u\in {\mathcal {U}}}"></span> such that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{t}u+u\partial _{x}u=\rho \partial _{xx}u+f\quad \forall x\in \left[0,2\pi \right),\forall t>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mi>u</mi> <mo>+</mo> <mi>u</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>u</mi> <mo>=</mo> <mi>ρ</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mi>u</mi> <mo>+</mo> <mi>f</mi> <mspace width="1em" /> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>∈</mo> <mrow> <mo>[</mo> <mrow> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mi>π</mi> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mi mathvariant="normal">∀</mi> <mi>t</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{t}u+u\partial _{x}u=\rho \partial _{xx}u+f\quad \forall x\in \left[0,2\pi \right),\forall t>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0f0089fa0f063e144e70d33d2eaa65694929286" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.235ex; height:2.843ex;" alt="{\displaystyle \partial _{t}u+u\partial _{x}u=\rho \partial _{xx}u+f\quad \forall x\in \left[0,2\pi \right),\forall t>0}"></span></dd></dl> <p>where ρ is the <a href="/wiki/Viscosity" title="Viscosity">viscosity</a> coefficient. In weak conservative form this becomes </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\langle \partial _{t}u,v\right\rangle ={\Bigl \langle }\partial _{x}\left(-{\tfrac {1}{2}}u^{2}+\rho \partial _{x}u\right),v{\Bigr \rangle }+\left\langle f,v\right\rangle \quad \forall v\in {\mathcal {V}},\forall t>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⟨</mo> <mrow> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mi>u</mi> <mo>,</mo> <mi>v</mi> </mrow> <mo>⟩</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">⟨</mo> </mrow> </mrow> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>ρ</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>u</mi> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">⟩</mo> </mrow> </mrow> <mo>+</mo> <mrow> <mo>⟨</mo> <mrow> <mi>f</mi> <mo>,</mo> <mi>v</mi> </mrow> <mo>⟩</mo> </mrow> <mspace width="1em" /> <mi mathvariant="normal">∀</mi> <mi>v</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> <mo>,</mo> <mi mathvariant="normal">∀</mi> <mi>t</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle \partial _{t}u,v\right\rangle ={\Bigl \langle }\partial _{x}\left(-{\tfrac {1}{2}}u^{2}+\rho \partial _{x}u\right),v{\Bigr \rangle }+\left\langle f,v\right\rangle \quad \forall v\in {\mathcal {V}},\forall t>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/947491345ffb6b5a15f9eee9b617e55d9b6bf652" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:59.418ex; height:4.843ex;" alt="{\displaystyle \left\langle \partial _{t}u,v\right\rangle ={\Bigl \langle }\partial _{x}\left(-{\tfrac {1}{2}}u^{2}+\rho \partial _{x}u\right),v{\Bigr \rangle }+\left\langle f,v\right\rangle \quad \forall v\in {\mathcal {V}},\forall t>0}"></span></dd></dl> <p>where following <a href="/wiki/Inner_product_space" title="Inner product space">inner product</a> notation. <a href="/wiki/Integration_by_parts" title="Integration by parts">Integrating by parts</a> and using periodicity grants </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \partial _{t}u,v\rangle =\left\langle {\tfrac {1}{2}}u^{2}-\rho \partial _{x}u,\partial _{x}v\right\rangle +\left\langle f,v\right\rangle \quad \forall v\in {\mathcal {V}},\forall t>0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow> <mo>⟨</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>ρ</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>u</mi> <mo>,</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>v</mi> </mrow> <mo>⟩</mo> </mrow> <mo>+</mo> <mrow> <mo>⟨</mo> <mrow> <mi>f</mi> <mo>,</mo> <mi>v</mi> </mrow> <mo>⟩</mo> </mrow> <mspace width="1em" /> <mi mathvariant="normal">∀</mi> <mi>v</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> <mo>,</mo> <mi mathvariant="normal">∀</mi> <mi>t</mi> <mo>></mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \partial _{t}u,v\rangle =\left\langle {\tfrac {1}{2}}u^{2}-\rho \partial _{x}u,\partial _{x}v\right\rangle +\left\langle f,v\right\rangle \quad \forall v\in {\mathcal {V}},\forall t>0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/accbdf2518569b0a240498e42e5e3c5077da5843" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:54.707ex; height:3.509ex;" alt="{\displaystyle \langle \partial _{t}u,v\rangle =\left\langle {\tfrac {1}{2}}u^{2}-\rho \partial _{x}u,\partial _{x}v\right\rangle +\left\langle f,v\right\rangle \quad \forall v\in {\mathcal {V}},\forall t>0.}"></span></dd></dl> <p>To apply the Fourier–<a href="/wiki/Galerkin_method" title="Galerkin method">Galerkin method</a>, choose both </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {U}}^{N}:={\biggl \{}u:u(x,t)=\sum _{k=-N/2}^{N/2-1}{\hat {u}}_{k}(t)e^{ikx}{\biggr \}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">U</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msup> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">{</mo> </mrow> </mrow> <mi>u</mi> <mo>:</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mo>−</mo> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>x</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">}</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {U}}^{N}:={\biggl \{}u:u(x,t)=\sum _{k=-N/2}^{N/2-1}{\hat {u}}_{k}(t)e^{ikx}{\biggr \}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e87affeade085719cd0f145e340a984736a7d259" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; margin-left: -0.038ex; width:38.959ex; height:8.176ex;" alt="{\displaystyle {\mathcal {U}}^{N}:={\biggl \{}u:u(x,t)=\sum _{k=-N/2}^{N/2-1}{\hat {u}}_{k}(t)e^{ikx}{\biggr \}}}"></span></dd></dl> <p>and </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {V}}^{N}:=\operatorname {span} \left\{e^{ikx}:k\in -{\tfrac {1}{2}}N,\dots ,{\tfrac {1}{2}}N-1\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msup> <mo>:=</mo> <mi>span</mi> <mo>⁡</mo> <mrow> <mo>{</mo> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>x</mi> </mrow> </msup> <mo>:</mo> <mi>k</mi> <mo>∈</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>N</mi> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {V}}^{N}:=\operatorname {span} \left\{e^{ikx}:k\in -{\tfrac {1}{2}}N,\dots ,{\tfrac {1}{2}}N-1\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ba60769c36120071d072dc85db7d1b0095e4c5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:42.473ex; height:3.509ex;" alt="{\displaystyle {\mathcal {V}}^{N}:=\operatorname {span} \left\{e^{ikx}:k\in -{\tfrac {1}{2}}N,\dots ,{\tfrac {1}{2}}N-1\right\}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {u}}_{k}(t):={\frac {1}{2\pi }}\langle u(x,t),e^{ikx}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>x</mi> </mrow> </msup> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {u}}_{k}(t):={\frac {1}{2\pi }}\langle u(x,t),e^{ikx}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79d65a4b851769c538da175786ecc4c1450bc763" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.009ex; height:5.176ex;" alt="{\displaystyle {\hat {u}}_{k}(t):={\frac {1}{2\pi }}\langle u(x,t),e^{ikx}\rangle }"></span>. This reduces the problem to finding <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\in {\mathcal {U}}^{N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">U</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u\in {\mathcal {U}}^{N}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b214f3bbca702ba028e870afdca66868696e08b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.501ex; height:2.676ex;" alt="{\displaystyle u\in {\mathcal {U}}^{N}}"></span> such that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \partial _{t}u,e^{ikx}\rangle =\left\langle {\tfrac {1}{2}}u^{2}-\rho \partial _{x}u,\partial _{x}e^{ikx}\right\rangle +\left\langle f,e^{ikx}\right\rangle \quad \forall k\in \left\{-{\tfrac {1}{2}}N,\dots ,{\tfrac {1}{2}}N-1\right\},\forall t>0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mi>u</mi> <mo>,</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>x</mi> </mrow> </msup> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow> <mo>⟨</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>ρ</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>u</mi> <mo>,</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>x</mi> </mrow> </msup> </mrow> <mo>⟩</mo> </mrow> <mo>+</mo> <mrow> <mo>⟨</mo> <mrow> <mi>f</mi> <mo>,</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>x</mi> </mrow> </msup> </mrow> <mo>⟩</mo> </mrow> <mspace width="1em" /> <mi mathvariant="normal">∀</mi> <mi>k</mi> <mo>∈</mo> <mrow> <mo>{</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>N</mi> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>}</mo> </mrow> <mo>,</mo> <mi mathvariant="normal">∀</mi> <mi>t</mi> <mo>></mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \partial _{t}u,e^{ikx}\rangle =\left\langle {\tfrac {1}{2}}u^{2}-\rho \partial _{x}u,\partial _{x}e^{ikx}\right\rangle +\left\langle f,e^{ikx}\right\rangle \quad \forall k\in \left\{-{\tfrac {1}{2}}N,\dots ,{\tfrac {1}{2}}N-1\right\},\forall t>0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38c2de13933f2892a31b7ebd5189a9c97dda6db0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:82.834ex; height:3.509ex;" alt="{\displaystyle \langle \partial _{t}u,e^{ikx}\rangle =\left\langle {\tfrac {1}{2}}u^{2}-\rho \partial _{x}u,\partial _{x}e^{ikx}\right\rangle +\left\langle f,e^{ikx}\right\rangle \quad \forall k\in \left\{-{\tfrac {1}{2}}N,\dots ,{\tfrac {1}{2}}N-1\right\},\forall t>0.}"></span></dd></dl> <p>Using the <a href="/wiki/Orthogonality" title="Orthogonality">orthogonality</a> relation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle e^{ilx},e^{ikx}\rangle =2\pi \delta _{lk}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>l</mi> <mi>x</mi> </mrow> </msup> <mo>,</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>x</mi> </mrow> </msup> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mn>2</mn> <mi>π</mi> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle e^{ilx},e^{ikx}\rangle =2\pi \delta _{lk}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a354905a55cee44778359f90fc8bd5e660ebe93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.041ex; height:3.176ex;" alt="{\displaystyle \langle e^{ilx},e^{ikx}\rangle =2\pi \delta _{lk}}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{lk}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{lk}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7db290f991897b9c03b7cbc46b5a67e56056d58a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.611ex; height:2.676ex;" alt="{\displaystyle \delta _{lk}}"></span> is the <a href="/wiki/Kronecker_delta" title="Kronecker delta">Kronecker delta</a>, we simplify the above three terms for each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> to see </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\left\langle \partial _{t}u,e^{ikx}\right\rangle &={\biggl \langle }\partial _{t}\sum _{l}{\hat {u}}_{l}e^{ilx},e^{ikx}{\biggr \rangle }={\biggl \langle }\sum _{l}\partial _{t}{\hat {u}}_{l}e^{ilx},e^{ikx}{\biggr \rangle }=2\pi \partial _{t}{\hat {u}}_{k},\\\left\langle f,e^{ikx}\right\rangle &={\biggl \langle }\sum _{l}{\hat {f}}_{l}e^{ilx},e^{ikx}{\biggr \rangle }=2\pi {\hat {f}}_{k},{\text{ and}}\\\left\langle {\tfrac {1}{2}}u^{2}-\rho \partial _{x}u,\partial _{x}e^{ikx}\right\rangle &={\biggl \langle }{\tfrac {1}{2}}{\Bigl (}\sum _{p}{\hat {u}}_{p}e^{ipx}{\Bigr )}{\Bigl (}\sum _{q}{\hat {u}}_{q}e^{iqx}{\Bigr )}-\rho \partial _{x}\sum _{l}{\hat {u}}_{l}e^{ilx},\partial _{x}e^{ikx}{\biggr \rangle }\\&={\biggl \langle }{\tfrac {1}{2}}\sum _{p}\sum _{q}{\hat {u}}_{p}{\hat {u}}_{q}e^{i\left(p+q\right)x},ike^{ikx}{\biggr \rangle }-{\biggl \langle }\rho i\sum _{l}l{\hat {u}}_{l}e^{ilx},ike^{ikx}{\biggr \rangle }\\&=-{\tfrac {1}{2}}ik{\biggl \langle }\sum _{p}\sum _{q}{\hat {u}}_{p}{\hat {u}}_{q}e^{i\left(p+q\right)x},e^{ikx}{\biggr \rangle }-\rho k{\biggl \langle }\sum _{l}l{\hat {u}}_{l}e^{ilx},e^{ikx}{\biggr \rangle }\\&=-i\pi k\sum _{p+q=k}{\hat {u}}_{p}{\hat {u}}_{q}-2\pi \rho {}k^{2}{\hat {u}}_{k}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow> <mo>⟨</mo> <mrow> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mi>u</mi> <mo>,</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>x</mi> </mrow> </msup> </mrow> <mo>⟩</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">⟨</mo> </mrow> </mrow> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </munder> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>l</mi> <mi>x</mi> </mrow> </msup> <mo>,</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>x</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">⟩</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">⟨</mo> </mrow> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </munder> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>l</mi> <mi>x</mi> </mrow> </msup> <mo>,</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>x</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">⟩</mo> </mrow> </mrow> <mo>=</mo> <mn>2</mn> <mi>π</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>⟨</mo> <mrow> <mi>f</mi> <mo>,</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>x</mi> </mrow> </msup> </mrow> <mo>⟩</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">⟨</mo> </mrow> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </munder> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>l</mi> <mi>x</mi> </mrow> </msup> <mo>,</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>x</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">⟩</mo> </mrow> </mrow> <mo>=</mo> <mn>2</mn> <mi>π</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> and</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>⟨</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>ρ</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>u</mi> <mo>,</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>x</mi> </mrow> </msup> </mrow> <mo>⟩</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">⟨</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </munder> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>p</mi> <mi>x</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </munder> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>q</mi> <mi>x</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mo>−</mo> <mi>ρ</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </munder> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>l</mi> <mi>x</mi> </mrow> </msup> <mo>,</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>x</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">⟩</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">⟨</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </munder> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </munder> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>+</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> <mi>x</mi> </mrow> </msup> <mo>,</mo> <mi>i</mi> <mi>k</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>x</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">⟩</mo> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">⟨</mo> </mrow> </mrow> <mi>ρ</mi> <mi>i</mi> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </munder> <mi>l</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>l</mi> <mi>x</mi> </mrow> </msup> <mo>,</mo> <mi>i</mi> <mi>k</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>x</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">⟩</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>i</mi> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">⟨</mo> </mrow> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </munder> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </munder> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>+</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> <mi>x</mi> </mrow> </msup> <mo>,</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>x</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">⟩</mo> </mrow> </mrow> <mo>−</mo> <mi>ρ</mi> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">⟨</mo> </mrow> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </munder> <mi>l</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>l</mi> <mi>x</mi> </mrow> </msup> <mo>,</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>x</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">⟩</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mi>π</mi> <mi>k</mi> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>+</mo> <mi>q</mi> <mo>=</mo> <mi>k</mi> </mrow> </munder> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo>−</mo> <mn>2</mn> <mi>π</mi> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\left\langle \partial _{t}u,e^{ikx}\right\rangle &={\biggl \langle }\partial _{t}\sum _{l}{\hat {u}}_{l}e^{ilx},e^{ikx}{\biggr \rangle }={\biggl \langle }\sum _{l}\partial _{t}{\hat {u}}_{l}e^{ilx},e^{ikx}{\biggr \rangle }=2\pi \partial _{t}{\hat {u}}_{k},\\\left\langle f,e^{ikx}\right\rangle &={\biggl \langle }\sum _{l}{\hat {f}}_{l}e^{ilx},e^{ikx}{\biggr \rangle }=2\pi {\hat {f}}_{k},{\text{ and}}\\\left\langle {\tfrac {1}{2}}u^{2}-\rho \partial _{x}u,\partial _{x}e^{ikx}\right\rangle &={\biggl \langle }{\tfrac {1}{2}}{\Bigl (}\sum _{p}{\hat {u}}_{p}e^{ipx}{\Bigr )}{\Bigl (}\sum _{q}{\hat {u}}_{q}e^{iqx}{\Bigr )}-\rho \partial _{x}\sum _{l}{\hat {u}}_{l}e^{ilx},\partial _{x}e^{ikx}{\biggr \rangle }\\&={\biggl \langle }{\tfrac {1}{2}}\sum _{p}\sum _{q}{\hat {u}}_{p}{\hat {u}}_{q}e^{i\left(p+q\right)x},ike^{ikx}{\biggr \rangle }-{\biggl \langle }\rho i\sum _{l}l{\hat {u}}_{l}e^{ilx},ike^{ikx}{\biggr \rangle }\\&=-{\tfrac {1}{2}}ik{\biggl \langle }\sum _{p}\sum _{q}{\hat {u}}_{p}{\hat {u}}_{q}e^{i\left(p+q\right)x},e^{ikx}{\biggr \rangle }-\rho k{\biggl \langle }\sum _{l}l{\hat {u}}_{l}e^{ilx},e^{ikx}{\biggr \rangle }\\&=-i\pi k\sum _{p+q=k}{\hat {u}}_{p}{\hat {u}}_{q}-2\pi \rho {}k^{2}{\hat {u}}_{k}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a83dc4b2912216e1200ec38bebbd5966cc00b0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -19.838ex; width:81.851ex; height:40.843ex;" alt="{\displaystyle {\begin{aligned}\left\langle \partial _{t}u,e^{ikx}\right\rangle &={\biggl \langle }\partial _{t}\sum _{l}{\hat {u}}_{l}e^{ilx},e^{ikx}{\biggr \rangle }={\biggl \langle }\sum _{l}\partial _{t}{\hat {u}}_{l}e^{ilx},e^{ikx}{\biggr \rangle }=2\pi \partial _{t}{\hat {u}}_{k},\\\left\langle f,e^{ikx}\right\rangle &={\biggl \langle }\sum _{l}{\hat {f}}_{l}e^{ilx},e^{ikx}{\biggr \rangle }=2\pi {\hat {f}}_{k},{\text{ and}}\\\left\langle {\tfrac {1}{2}}u^{2}-\rho \partial _{x}u,\partial _{x}e^{ikx}\right\rangle &={\biggl \langle }{\tfrac {1}{2}}{\Bigl (}\sum _{p}{\hat {u}}_{p}e^{ipx}{\Bigr )}{\Bigl (}\sum _{q}{\hat {u}}_{q}e^{iqx}{\Bigr )}-\rho \partial _{x}\sum _{l}{\hat {u}}_{l}e^{ilx},\partial _{x}e^{ikx}{\biggr \rangle }\\&={\biggl \langle }{\tfrac {1}{2}}\sum _{p}\sum _{q}{\hat {u}}_{p}{\hat {u}}_{q}e^{i\left(p+q\right)x},ike^{ikx}{\biggr \rangle }-{\biggl \langle }\rho i\sum _{l}l{\hat {u}}_{l}e^{ilx},ike^{ikx}{\biggr \rangle }\\&=-{\tfrac {1}{2}}ik{\biggl \langle }\sum _{p}\sum _{q}{\hat {u}}_{p}{\hat {u}}_{q}e^{i\left(p+q\right)x},e^{ikx}{\biggr \rangle }-\rho k{\biggl \langle }\sum _{l}l{\hat {u}}_{l}e^{ilx},e^{ikx}{\biggr \rangle }\\&=-i\pi k\sum _{p+q=k}{\hat {u}}_{p}{\hat {u}}_{q}-2\pi \rho {}k^{2}{\hat {u}}_{k}.\end{aligned}}}"></span></dd></dl> <p>Assemble the three terms for each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> to obtain </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi \partial _{t}{\hat {u}}_{k}=-i\pi k\sum _{p+q=k}{\hat {u}}_{p}{\hat {u}}_{q}-2\pi \rho {}k^{2}{\hat {u}}_{k}+2\pi {\hat {f}}_{k}\quad k\in \left\{-{\tfrac {1}{2}}N,\dots ,{\tfrac {1}{2}}N-1\right\},\forall t>0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mi>π</mi> <mi>k</mi> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>+</mo> <mi>q</mi> <mo>=</mo> <mi>k</mi> </mrow> </munder> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo>−</mo> <mn>2</mn> <mi>π</mi> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>+</mo> <mn>2</mn> <mi>π</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mspace width="1em" /> <mi>k</mi> <mo>∈</mo> <mrow> <mo>{</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>N</mi> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>}</mo> </mrow> <mo>,</mo> <mi mathvariant="normal">∀</mi> <mi>t</mi> <mo>></mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi \partial _{t}{\hat {u}}_{k}=-i\pi k\sum _{p+q=k}{\hat {u}}_{p}{\hat {u}}_{q}-2\pi \rho {}k^{2}{\hat {u}}_{k}+2\pi {\hat {f}}_{k}\quad k\in \left\{-{\tfrac {1}{2}}N,\dots ,{\tfrac {1}{2}}N-1\right\},\forall t>0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8146cdcdc1b69f11b07da2acdaa49b57cab3bf2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:81.025ex; height:5.843ex;" alt="{\displaystyle 2\pi \partial _{t}{\hat {u}}_{k}=-i\pi k\sum _{p+q=k}{\hat {u}}_{p}{\hat {u}}_{q}-2\pi \rho {}k^{2}{\hat {u}}_{k}+2\pi {\hat {f}}_{k}\quad k\in \left\{-{\tfrac {1}{2}}N,\dots ,{\tfrac {1}{2}}N-1\right\},\forall t>0.}"></span></dd></dl> <p>Dividing through by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }"></span>, we finally arrive at </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{t}{\hat {u}}_{k}=-{\frac {ik}{2}}\sum _{p+q=k}{\hat {u}}_{p}{\hat {u}}_{q}-\rho {}k^{2}{\hat {u}}_{k}+{\hat {f}}_{k}\quad k\in \left\{-{\tfrac {1}{2}}N,\dots ,{\tfrac {1}{2}}N-1\right\},\forall t>0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mi>k</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>+</mo> <mi>q</mi> <mo>=</mo> <mi>k</mi> </mrow> </munder> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo>−</mo> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mspace width="1em" /> <mi>k</mi> <mo>∈</mo> <mrow> <mo>{</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>N</mi> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>}</mo> </mrow> <mo>,</mo> <mi mathvariant="normal">∀</mi> <mi>t</mi> <mo>></mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{t}{\hat {u}}_{k}=-{\frac {ik}{2}}\sum _{p+q=k}{\hat {u}}_{p}{\hat {u}}_{q}-\rho {}k^{2}{\hat {u}}_{k}+{\hat {f}}_{k}\quad k\in \left\{-{\tfrac {1}{2}}N,\dots ,{\tfrac {1}{2}}N-1\right\},\forall t>0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1209263e9f58b1b9db8f9a6de78a6eb4c5b4ed6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:73.046ex; height:6.843ex;" alt="{\displaystyle \partial _{t}{\hat {u}}_{k}=-{\frac {ik}{2}}\sum _{p+q=k}{\hat {u}}_{p}{\hat {u}}_{q}-\rho {}k^{2}{\hat {u}}_{k}+{\hat {f}}_{k}\quad k\in \left\{-{\tfrac {1}{2}}N,\dots ,{\tfrac {1}{2}}N-1\right\},\forall t>0.}"></span></dd></dl> <p>With Fourier transformed initial conditions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {u}}_{k}(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {u}}_{k}(0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8eff26f91b40e5722df02f52e91e994b49f0e623" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.39ex; height:2.843ex;" alt="{\displaystyle {\hat {u}}_{k}(0)}"></span> and forcing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}_{k}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}_{k}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdb6c56251b1d693c24f70efa24fe8c8d2a11bfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.437ex; height:3.343ex;" alt="{\displaystyle {\hat {f}}_{k}(t)}"></span>, this coupled system of ordinary differential equations may be integrated in time (using, e.g., a <a href="/wiki/Runge_Kutta" class="mw-redirect" title="Runge Kutta">Runge Kutta</a> technique) to find a solution. The nonlinear term is a <a href="/wiki/Convolution" title="Convolution">convolution</a>, and there are several transform-based techniques for evaluating it efficiently. See the references by Boyd and Canuto et al. for more details. </p> <div class="mw-heading mw-heading2"><h2 id="A_relationship_with_the_spectral_element_method">A relationship with the spectral element method</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spectral_method&action=edit&section=5" title="Edit section: A relationship with the spectral element method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One can show that if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span> is infinitely differentiable, then the numerical algorithm using Fast Fourier Transforms will converge faster than any polynomial in the grid size h. That is, for any n>0, there is a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{n}<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{n}<\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d66ab48a0b9ab73915f9ce4c5c34ff929557462" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.302ex; height:2.509ex;" alt="{\displaystyle C_{n}<\infty }"></span> such that the error is less than <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{n}h^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{n}h^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0877af32c01f1e03d1fc5c161abc01294f6a06ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.438ex; height:2.676ex;" alt="{\displaystyle C_{n}h^{n}}"></span> for all sufficiently small values of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span>. We say that the spectral method is of order <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, for every n>0. </p><p>Because a <a href="/wiki/Spectral_element_method" title="Spectral element method">spectral element method</a> is a <a href="/wiki/Finite_element_method" title="Finite element method">finite element method</a> of very high order, there is a similarity in the convergence properties. However, whereas the spectral method is based on the eigendecomposition of the particular boundary value problem, the finite element method does not use that information and works for arbitrary <a href="/wiki/Elliptic_boundary_value_problem" title="Elliptic boundary value problem">elliptic boundary value problems</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spectral_method&action=edit&section=6" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Finite_element_method" title="Finite element method">Finite element method</a></li> <li><a href="/wiki/Gaussian_grid" title="Gaussian grid">Gaussian grid</a></li> <li><a href="/wiki/Pseudo-spectral_method" title="Pseudo-spectral method">Pseudo-spectral method</a></li> <li><a href="/wiki/Spectral_element_method" title="Spectral element method">Spectral element method</a></li> <li><a href="/wiki/Galerkin_method" title="Galerkin method">Galerkin method</a></li> <li><a href="/wiki/Collocation_method" title="Collocation method">Collocation method</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spectral_method&action=edit&section=7" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.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}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-CHQZ-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-CHQZ_1-0">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://books.google.com/books?id=7COgEw5_EBQC">pp 235, Spectral Methods</a>: evolution to complex geometries and applications to fluid dynamics, By Canuto, Hussaini, Quarteroni and Zang, Springer, 2007.</span> </li> </ol></div></div> <ul><li>Bengt Fornberg (1996) <i>A Practical Guide to Pseudospectral Methods.</i> Cambridge University Press, Cambridge, UK</li> <li><a rel="nofollow" class="external text" href="http://www-personal.umich.edu/~jpboyd/BOOK_Spectral2000.html">Chebyshev and Fourier Spectral Methods</a> by John P. Boyd.</li> <li>Canuto C., <a href="/wiki/M._Yousuff_Hussaini" title="M. Yousuff Hussaini">Hussaini M. Y.</a>, Quarteroni A., and Zang T.A. (2006) <i>Spectral Methods. Fundamentals in Single Domains.</i> Springer-Verlag, Berlin Heidelberg</li> <li>Javier de Frutos, Julia Novo (2000): <a rel="nofollow" class="external text" href="https://epubs.siam.org/doi/10.1137/S0036142999351984">A Spectral Element Method for the Navier–Stokes Equations with Improved Accuracy</a></li> <li><a rel="nofollow" class="external text" href="http://cdm.unimo.it/home/matematica/funaro.daniele/bube.htm">Polynomial Approximation of Differential Equations</a>, by Daniele Funaro, Lecture Notes in Physics, Volume 8, Springer-Verlag, Heidelberg 1992</li> <li>D. Gottlieb and S. Orzag (1977) "Numerical Analysis of Spectral Methods : Theory and Applications", SIAM, Philadelphia, PA</li> <li>J. Hesthaven, S. Gottlieb and D. Gottlieb (2007) "Spectral methods for time-dependent problems", Cambridge UP, Cambridge, UK</li> <li>Steven A. Orszag (1969) <i>Numerical Methods for the Simulation of Turbulence</i>, Phys. Fluids Supp. II, 12, 250–257</li> <li><style data-mw-deduplicate="TemplateStyles:r1238218222">.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("//upload.wikimedia.org/wikipedia/commons/6/65/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("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/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}}</style><cite id="CITEREFPressTeukolskyVetterlingFlannery2007" class="citation book cs1">Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). <a rel="nofollow" class="external text" href="http://apps.nrbook.com/empanel/index.html#pg=1083">"Section 20.7. Spectral Methods"</a>. <i>Numerical Recipes: The Art of Scientific Computing</i> (3rd ed.). New York: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-88068-8" title="Special:BookSources/978-0-521-88068-8"><bdi>978-0-521-88068-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Section+20.7.+Spectral+Methods&rft.btitle=Numerical+Recipes%3A+The+Art+of+Scientific+Computing&rft.place=New+York&rft.edition=3rd&rft.pub=Cambridge+University+Press&rft.date=2007&rft.isbn=978-0-521-88068-8&rft.aulast=Press&rft.aufirst=WH&rft.au=Teukolsky%2C+SA&rft.au=Vetterling%2C+WT&rft.au=Flannery%2C+BP&rft_id=http%3A%2F%2Fapps.nrbook.com%2Fempanel%2Findex.html%23pg%3D1083&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpectral+method" class="Z3988"></span></li> <li>Jie Shen, Tao Tang and Li-Lian Wang (2011) "Spectral Methods: Algorithms, Analysis and Applications" (Springer Series in Computational Mathematics, V. 41, Springer), <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/354071040X" title="Special:BookSources/354071040X">354071040X</a></li> <li>Lloyd N. Trefethen (2000) <i>Spectral Methods in MATLAB.</i> SIAM, Philadelphia, PA</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.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:" · 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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}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Numerical_PDE" title="Template:Numerical PDE"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Numerical_PDE" title="Template talk:Numerical PDE"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Numerical_PDE" title="Special:EditPage/Template:Numerical PDE"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Numerical_methods_for_partial_differential_equations" style="font-size:114%;margin:0 4em"><a href="/wiki/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="/wiki/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="/wiki/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="/wiki/FTCS_scheme" title="FTCS scheme">Forward-time central-space</a> (FTCS)</li> <li><a href="/wiki/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="/wiki/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="/wiki/Lax%E2%80%93Friedrichs_method" title="Lax–Friedrichs method">Lax–Friedrichs</a></li> <li><a href="/wiki/Lax%E2%80%93Wendroff_method" title="Lax–Wendroff method">Lax–Wendroff</a></li> <li><a href="/wiki/MacCormack_method" title="MacCormack method">MacCormack</a></li> <li><a href="/wiki/Upwind_scheme" title="Upwind scheme">Upwind</a></li> <li><a href="/wiki/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="/wiki/Alternating_direction_implicit_method" class="mw-redirect" title="Alternating direction implicit method">Alternating direction-implicit</a> (ADI)</li> <li><a href="/wiki/Finite-difference_frequency-domain_method" title="Finite-difference frequency-domain method">Finite-difference frequency-domain</a> (FDFD)</li> <li><a href="/wiki/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="/wiki/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="/wiki/Godunov%27s_scheme" title="Godunov's scheme">Godunov</a></li> <li><a href="/wiki/High-resolution_scheme" title="High-resolution scheme">High-resolution</a></li> <li><a href="/wiki/MUSCL_scheme" title="MUSCL scheme">Monotonic upstream-centered</a> (MUSCL)</li> <li><a href="/wiki/AUSM" class="mw-redirect" title="AUSM">Advection upstream-splitting</a> (AUSM)</li> <li><a href="/wiki/Riemann_solver" title="Riemann solver">Riemann solver</a></li> <li><a href="/wiki/ENO_methods" title="ENO methods">Essentially non-oscillatory</a> (ENO)</li> <li><a href="/wiki/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="/wiki/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"> <ul><li><a href="/wiki/Hp-FEM" title="Hp-FEM">hp-FEM</a></li> <li><a href="/wiki/Extended_finite_element_method" title="Extended finite element method">Extended</a> (XFEM)</li> <li><a href="/wiki/Discontinuous_Galerkin_method" title="Discontinuous Galerkin method">Discontinuous Galerkin</a> (DG)</li> <li><a href="/wiki/Spectral_element_method" title="Spectral element method">Spectral element</a> (SEM)</li> <li><a href="/wiki/Mortar_methods" title="Mortar methods">Mortar</a></li> <li><a href="/wiki/Gradient_discretisation_method" title="Gradient discretisation method">Gradient discretisation</a> (GDM)</li> <li><a href="/wiki/Loubignac_iteration" title="Loubignac iteration">Loubignac iteration</a></li> <li><a href="/wiki/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="/wiki/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="/wiki/Smoothed-particle_hydrodynamics" title="Smoothed-particle hydrodynamics">Smoothed-particle hydrodynamics</a> (SPH)</li> <li><a href="/wiki/Peridynamics" title="Peridynamics">Peridynamics</a> (PD)</li> <li><a href="/wiki/Moving_particle_semi-implicit_method" title="Moving particle semi-implicit method">Moving particle semi-implicit method</a> (MPS)</li> <li><a href="/wiki/Material_point_method" title="Material point method">Material point method</a> (MPM)</li> <li><a href="/wiki/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="/wiki/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="/wiki/Schur_complement_method" title="Schur complement method">Schur complement</a></li> <li><a href="/wiki/Fictitious_domain_method" title="Fictitious domain method">Fictitious domain</a></li> <li><a href="/wiki/Schwarz_alternating_method" title="Schwarz alternating method">Schwarz alternating</a> <ul><li><a href="/wiki/Additive_Schwarz_method" title="Additive Schwarz method">additive</a></li> <li><a href="/wiki/Abstract_additive_Schwarz_method" title="Abstract additive Schwarz method">abstract additive</a></li></ul></li> <li><a href="/wiki/Neumann%E2%80%93Dirichlet_method" title="Neumann–Dirichlet method">Neumann–Dirichlet</a></li> <li><a href="/wiki/Neumann%E2%80%93Neumann_methods" title="Neumann–Neumann methods">Neumann–Neumann</a></li> <li><a href="/wiki/Poincar%C3%A9%E2%80%93Steklov_operator" title="Poincaré–Steklov operator">Poincaré–Steklov operator</a></li> <li><a href="/wiki/Balancing_domain_decomposition_method" title="Balancing domain decomposition method">Balancing</a> (BDD)</li> <li><a href="/wiki/BDDC" title="BDDC">Balancing by constraints</a> (BDDC)</li> <li><a href="/wiki/FETI" title="FETI">Tearing and interconnect</a> (FETI)</li> <li><a href="/wiki/FETI-DP" title="FETI-DP">FETI-DP</a></li></ul> </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 class="mw-selflink selflink">Spectral</a></li> <li><a href="/wiki/Pseudo-spectral_method" title="Pseudo-spectral method">Pseudospectral</a> (DVR)</li> <li><a href="/wiki/Method_of_lines" title="Method of lines">Method of lines</a></li> <li><a href="/wiki/Multigrid_method" title="Multigrid method">Multigrid</a></li> <li><a href="/wiki/Collocation_method" title="Collocation method">Collocation</a></li> <li><a href="/wiki/Level-set_method" title="Level-set method">Level-set</a></li> <li><a href="/wiki/Boundary_element_method" title="Boundary element method">Boundary element</a> <ul><li><a href="/wiki/Method_of_moments_(electromagnetics)" title="Method of moments (electromagnetics)">Method of moments</a></li></ul></li> <li><a href="/wiki/Immersed_boundary_method" title="Immersed boundary method">Immersed boundary</a></li> <li><a href="/wiki/Analytic_element_method" title="Analytic element method">Analytic element</a></li> <li><a href="/wiki/Isogeometric_analysis" title="Isogeometric analysis">Isogeometric analysis</a></li> <li><a href="/wiki/Infinite_difference_method" title="Infinite difference method">Infinite difference method</a></li> <li><a href="/wiki/Infinite_element_method" title="Infinite element method">Infinite element method</a></li> <li><a href="/wiki/Galerkin_method" title="Galerkin method">Galerkin method</a> <ul><li><a href="/wiki/Petrov%E2%80%93Galerkin_method" title="Petrov–Galerkin method">Petrov–Galerkin method</a></li></ul></li> <li><a href="/wiki/Validated_numerics" title="Validated numerics">Validated numerics</a></li> <li><a href="/wiki/Computer-assisted_proof" title="Computer-assisted proof">Computer-assisted proof</a></li> <li><a href="/wiki/Integrable_algorithm" title="Integrable algorithm">Integrable algorithm</a></li> <li><a href="/wiki/Method_of_fundamental_solutions" title="Method of fundamental solutions">Method of fundamental solutions</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Functional_analysis_(topics_–_glossary)" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Functional_analysis" title="Template:Functional analysis"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Functional_analysis" title="Template talk:Functional analysis"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Functional_analysis" title="Special:EditPage/Template:Functional analysis"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Functional_analysis_(topics_–_glossary)" style="font-size:114%;margin:0 4em"><a href="/wiki/Functional_analysis" title="Functional analysis">Functional analysis</a> (<a href="/wiki/List_of_functional_analysis_topics" title="List of functional analysis topics">topics</a> – <a href="/wiki/Glossary_of_functional_analysis" title="Glossary of functional analysis">glossary</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Spaces</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Banach_space" title="Banach space">Banach</a></li> <li><a href="/wiki/Besov_space" title="Besov space">Besov</a></li> <li><a href="/wiki/Fr%C3%A9chet_space" title="Fréchet space">Fréchet</a></li> <li><a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert</a></li> <li><a href="/wiki/H%C3%B6lder_space" class="mw-redirect" title="Hölder space">Hölder</a></li> <li><a href="/wiki/Nuclear_space" title="Nuclear space">Nuclear</a></li> <li><a href="/wiki/Orlicz_space" title="Orlicz space">Orlicz</a></li> <li><a href="/wiki/Schwartz_space" title="Schwartz space">Schwartz</a></li> <li><a href="/wiki/Sobolev_space" title="Sobolev space">Sobolev</a></li> <li><a href="/wiki/Topological_vector_space" title="Topological vector space">Topological vector</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties</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="/wiki/Barrelled_space" title="Barrelled space">Barrelled</a></li> <li><a href="/wiki/Complete_topological_vector_space" title="Complete topological vector space">Complete</a></li> <li><a href="/wiki/Dual_space" title="Dual space">Dual</a> (<a href="/wiki/Dual_space#Algebraic_dual_space" title="Dual space">Algebraic</a>/<a href="/wiki/Dual_space#Continuous_dual_space" title="Dual space">Topological</a>)</li> <li><a href="/wiki/Locally_convex_topological_vector_space" title="Locally convex topological vector space">Locally convex</a></li> <li><a href="/wiki/Reflexive_space" title="Reflexive space">Reflexive</a></li> <li><a href="/wiki/Separable_space" title="Separable space">Separable</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Theorems_in_functional_analysis" title="Category:Theorems in functional analysis">Theorems</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="/wiki/Hahn%E2%80%93Banach_theorem" title="Hahn–Banach theorem">Hahn–Banach</a></li> <li><a href="/wiki/Riesz_representation_theorem" title="Riesz representation theorem">Riesz representation</a></li> <li><a href="/wiki/Closed_graph_theorem_(functional_analysis)" title="Closed graph theorem (functional analysis)">Closed graph</a></li> <li><a href="/wiki/Uniform_boundedness_principle" title="Uniform boundedness principle">Uniform boundedness principle</a></li> <li><a href="/wiki/Kakutani_fixed-point_theorem#Infinite-dimensional_generalizations" title="Kakutani fixed-point theorem">Kakutani fixed-point</a></li> <li><a href="/wiki/Krein%E2%80%93Milman_theorem" title="Krein–Milman theorem">Krein–Milman</a></li> <li><a href="/wiki/Min-max_theorem" title="Min-max theorem">Min–max</a></li> <li><a href="/wiki/Gelfand%E2%80%93Naimark_theorem" title="Gelfand–Naimark theorem">Gelfand–Naimark</a></li> <li><a href="/wiki/Banach%E2%80%93Alaoglu_theorem" title="Banach–Alaoglu theorem">Banach–Alaoglu</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Operators</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="/wiki/Adjoint_operator" class="mw-redirect" title="Adjoint operator">Adjoint</a></li> <li><a href="/wiki/Bounded_operator" title="Bounded operator">Bounded</a></li> <li><a href="/wiki/Compact_operator" title="Compact operator">Compact</a></li> <li><a href="/wiki/Hilbert%E2%80%93Schmidt_operator" title="Hilbert–Schmidt operator">Hilbert–Schmidt</a></li> <li><a href="/wiki/Normal_operator" title="Normal operator">Normal</a></li> <li><a href="/wiki/Nuclear_operator" title="Nuclear operator">Nuclear</a></li> <li><a href="/wiki/Trace_class" title="Trace class">Trace class</a></li> <li><a href="/wiki/Transpose_of_a_linear_map" title="Transpose of a linear map">Transpose</a></li> <li><a href="/wiki/Unbounded_operator" title="Unbounded operator">Unbounded</a></li> <li><a href="/wiki/Unitary_operator" title="Unitary operator">Unitary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Algebras</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="/wiki/Banach_algebra" title="Banach algebra">Banach algebra</a></li> <li><a href="/wiki/C*-algebra" title="C*-algebra">C*-algebra</a></li> <li><a href="/wiki/Spectrum_of_a_C*-algebra" title="Spectrum of a C*-algebra">Spectrum of a C*-algebra</a></li> <li><a href="/wiki/Operator_algebra" title="Operator algebra">Operator algebra</a></li> <li><a href="/wiki/Group_algebra_of_a_locally_compact_group" title="Group algebra of a locally compact group">Group algebra of a locally compact group</a></li> <li><a href="/wiki/Von_Neumann_algebra" title="Von Neumann algebra">Von Neumann algebra</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Open problems</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="/wiki/Invariant_subspace_problem" title="Invariant subspace problem">Invariant subspace problem</a></li> <li><a href="/wiki/Mahler%27s_conjecture" class="mw-redirect" title="Mahler's conjecture">Mahler's conjecture</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</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="/wiki/Hardy_space" title="Hardy space">Hardy space</a></li> <li><a href="/wiki/Spectral_theory_of_ordinary_differential_equations" title="Spectral theory of ordinary differential equations">Spectral theory of ordinary differential equations</a></li> <li><a href="/wiki/Heat_kernel" title="Heat kernel">Heat kernel</a></li> <li><a href="/wiki/Index_theorem" class="mw-redirect" title="Index theorem">Index theorem</a></li> <li><a href="/wiki/Calculus_of_variations" title="Calculus of variations">Calculus of variations</a></li> <li><a href="/wiki/Functional_calculus" title="Functional calculus">Functional calculus</a></li> <li><a href="/wiki/Integral_operator" title="Integral operator">Integral operator</a></li> <li><a href="/wiki/Jones_polynomial" title="Jones polynomial">Jones polynomial</a></li> <li><a href="/wiki/Topological_quantum_field_theory" title="Topological quantum field theory">Topological quantum field theory</a></li> <li><a href="/wiki/Noncommutative_geometry" title="Noncommutative geometry">Noncommutative geometry</a></li> <li><a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a></li> <li><a href="/wiki/Distribution_(mathematics)" title="Distribution (mathematics)">Distribution</a> (or <a href="/wiki/Generalized_function" title="Generalized function">Generalized functions</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Advanced topics</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="/wiki/Approximation_property" title="Approximation property">Approximation property</a></li> <li><a href="/wiki/Balanced_set" title="Balanced set">Balanced set</a></li> <li><a href="/wiki/Choquet_theory" title="Choquet theory">Choquet theory</a></li> <li><a href="/wiki/Weak_topology" title="Weak topology">Weak topology</a></li> <li><a href="/wiki/Banach%E2%80%93Mazur_distance" class="mw-redirect" title="Banach–Mazur distance">Banach–Mazur distance</a></li> <li><a href="/wiki/Tomita%E2%80%93Takesaki_theory" title="Tomita–Takesaki theory">Tomita–Takesaki theory</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Functional_analysis" title="Category:Functional analysis">Category</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Spectral_theory_and_*-algebras" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Spectral_theory" title="Template:Spectral theory"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Spectral_theory" title="Template talk:Spectral theory"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Spectral_theory" title="Special:EditPage/Template:Spectral theory"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Spectral_theory_and_*-algebras" style="font-size:114%;margin:0 4em"><a href="/wiki/Spectral_theory" title="Spectral theory">Spectral theory</a> and <a href="/wiki/*-algebra" title="*-algebra"><sup>*</sup>-algebras</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</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="/wiki/*-algebra" title="*-algebra">Involution/*-algebra</a></li> <li><a href="/wiki/Banach_algebra" title="Banach algebra">Banach algebra</a></li> <li><a href="/wiki/Banach_*-algebra" class="mw-redirect" title="Banach *-algebra">B*-algebra</a></li> <li><a href="/wiki/C*-algebra" title="C*-algebra">C*-algebra</a></li> <li><a href="/wiki/Noncommutative_topology" title="Noncommutative topology">Noncommutative topology</a></li> <li><a href="/wiki/Projection-valued_measure" title="Projection-valued measure">Projection-valued measure</a></li> <li><a href="/wiki/Spectrum_(functional_analysis)" title="Spectrum (functional analysis)">Spectrum</a></li> <li><a href="/wiki/Spectrum_of_a_C*-algebra" title="Spectrum of a C*-algebra">Spectrum of a C*-algebra</a></li> <li><a href="/wiki/Spectral_radius" title="Spectral radius">Spectral radius</a></li> <li><a href="/wiki/Operator_space" title="Operator space">Operator space</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Main results</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="/wiki/Gelfand%E2%80%93Mazur_theorem" title="Gelfand–Mazur theorem">Gelfand–Mazur theorem</a></li> <li><a href="/wiki/Gelfand%E2%80%93Naimark_theorem" title="Gelfand–Naimark theorem">Gelfand–Naimark theorem</a></li> <li><a href="/wiki/Gelfand_representation" title="Gelfand representation">Gelfand representation</a></li> <li><a href="/wiki/Polar_decomposition" title="Polar decomposition">Polar decomposition</a></li> <li><a href="/wiki/Singular_value_decomposition" title="Singular value decomposition">Singular value decomposition</a></li> <li><a href="/wiki/Spectral_theorem" title="Spectral theorem">Spectral theorem</a></li> <li><a href="/wiki/Spectral_theory_of_normal_C*-algebras" title="Spectral theory of normal C*-algebras">Spectral theory of normal C*-algebras</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Special Elements/Operators</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="/wiki/Isospectral" title="Isospectral">Isospectral</a></li> <li><a href="/wiki/Normal_element" title="Normal element">Normal</a> <a href="/wiki/Normal_operator" title="Normal operator">operator</a></li> <li><a href="/wiki/Self-adjoint" title="Self-adjoint">Hermitian/Self-adjoint</a> <a href="/wiki/Self-adjoint_operator" title="Self-adjoint operator">operator</a></li> <li><a href="/wiki/Unitary_element" title="Unitary element">Unitary</a> <a href="/wiki/Unitary_operator" title="Unitary operator">operator</a></li> <li><a href="/wiki/Unit_(ring_theory)" title="Unit (ring theory)">Unit</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Spectrum_(functional_analysis)" title="Spectrum (functional analysis)">Spectrum</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="/wiki/Krein%E2%80%93Rutman_theorem" title="Krein–Rutman theorem">Krein–Rutman theorem</a></li> <li><a href="/wiki/Normal_eigenvalue" title="Normal eigenvalue">Normal eigenvalue</a></li> <li><a href="/wiki/Spectrum_of_a_C*-algebra" title="Spectrum of a C*-algebra">Spectrum of a C*-algebra</a></li> <li><a href="/wiki/Spectral_radius" title="Spectral radius">Spectral radius</a></li> <li><a href="/wiki/Spectral_asymmetry" title="Spectral asymmetry">Spectral asymmetry</a></li> <li><a href="/wiki/Spectral_gap" title="Spectral gap">Spectral gap</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Decomposition</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="/wiki/Decomposition_of_spectrum_(functional_analysis)" title="Decomposition of spectrum (functional analysis)">Decomposition of a spectrum</a> <ul><li><a href="/wiki/Continuous_spectrum_(functional_analysis)" class="mw-redirect" title="Continuous spectrum (functional analysis)">Continuous</a></li> <li><a href="/wiki/Point_spectrum" class="mw-redirect" title="Point spectrum">Point</a></li> <li><a href="/wiki/Spectrum_(functional_analysis)#Residual_spectrum" title="Spectrum (functional analysis)">Residual</a></li></ul></li> <li><a href="/wiki/Spectrum_(functional_analysis)#Approximate_point_spectrum" title="Spectrum (functional analysis)">Approximate point</a></li> <li><a href="/wiki/Spectrum_(functional_analysis)#Compression_spectrum" title="Spectrum (functional analysis)">Compression</a></li> <li><a href="/wiki/Direct_integral" title="Direct integral">Direct integral</a></li> <li><a href="/wiki/Discrete_spectrum_(mathematics)" title="Discrete spectrum (mathematics)">Discrete</a></li> <li><a href="/wiki/Spectral_abscissa" title="Spectral abscissa">Spectral abscissa</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Spectral Theorem</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="/wiki/Borel_functional_calculus" title="Borel functional calculus">Borel functional calculus</a></li> <li><a href="/wiki/Min-max_theorem" title="Min-max theorem">Min-max theorem</a></li> <li><a href="/wiki/Positive_operator-valued_measure" class="mw-redirect" title="Positive operator-valued measure">Positive operator-valued measure</a></li> <li><a href="/wiki/Projection-valued_measure" title="Projection-valued measure">Projection-valued measure</a></li> <li><a href="/wiki/Riesz_projector" title="Riesz projector">Riesz projector</a></li> <li><a href="/wiki/Rigged_Hilbert_space" title="Rigged Hilbert space">Rigged Hilbert space</a></li> <li><a href="/wiki/Spectral_theorem" title="Spectral theorem">Spectral theorem</a></li> <li><a href="/wiki/Spectral_theory_of_compact_operators" title="Spectral theory of compact operators">Spectral theory of compact operators</a></li> <li><a href="/wiki/Spectral_theory_of_normal_C*-algebras" title="Spectral theory of normal C*-algebras">Spectral theory of normal C*-algebras</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Special algebras</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="/wiki/Amenable_Banach_algebra" title="Amenable Banach algebra">Amenable Banach algebra</a></li> <li>With an <a href="/wiki/Approximate_identity" title="Approximate identity">Approximate identity</a></li> <li><a href="/wiki/Banach_function_algebra" title="Banach function algebra">Banach function algebra</a></li> <li><a href="/wiki/Disk_algebra" title="Disk algebra">Disk algebra</a></li> <li><a href="/wiki/Nuclear_C*-algebra" title="Nuclear C*-algebra">Nuclear C*-algebra</a></li> <li><a href="/wiki/Uniform_algebra" title="Uniform algebra">Uniform algebra</a></li> <li><a href="/wiki/Von_Neumann_algebra" title="Von Neumann algebra">Von Neumann algebra</a> <ul><li><a href="/wiki/Tomita%E2%80%93Takesaki_theory" title="Tomita–Takesaki theory">Tomita–Takesaki theory</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Finite-Dimensional</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="/wiki/Alon%E2%80%93Boppana_bound" title="Alon–Boppana bound">Alon–Boppana bound</a></li> <li><a href="/wiki/Bauer%E2%80%93Fike_theorem" title="Bauer–Fike theorem">Bauer–Fike theorem</a></li> <li><a href="/wiki/Numerical_range" title="Numerical range">Numerical range</a></li> <li><a href="/wiki/Schur%E2%80%93Horn_theorem" title="Schur–Horn theorem">Schur–Horn theorem</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Generalizations</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="/wiki/Dirac_spectrum" title="Dirac spectrum">Dirac spectrum</a></li> <li><a href="/wiki/Essential_spectrum" title="Essential spectrum">Essential spectrum</a></li> <li><a href="/wiki/Pseudospectrum" title="Pseudospectrum">Pseudospectrum</a></li> <li><a href="/wiki/Structure_space" class="mw-redirect" title="Structure space">Structure space</a> (<a href="/wiki/Shilov_boundary" title="Shilov boundary">Shilov boundary</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Miscellaneous</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="/wiki/Abstract_index_group" class="mw-redirect" title="Abstract index group">Abstract index group</a></li> <li><a href="/wiki/Banach_algebra_cohomology" title="Banach algebra cohomology">Banach algebra cohomology</a></li> <li><a href="/wiki/Cohen%E2%80%93Hewitt_factorization_theorem" title="Cohen–Hewitt factorization theorem">Cohen–Hewitt factorization theorem</a></li> <li><a href="/wiki/Extensions_of_symmetric_operators" title="Extensions of symmetric operators">Extensions of symmetric operators</a></li> <li><a href="/wiki/Fredholm_theory" title="Fredholm theory">Fredholm theory</a></li> <li><a href="/wiki/Limiting_absorption_principle" title="Limiting absorption principle">Limiting absorption principle</a></li> <li><a href="/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorems_for_operator_algebras" title="Schröder–Bernstein theorems for operator algebras">Schröder–Bernstein theorems for operator algebras</a></li> <li><a href="/wiki/Sherman%E2%80%93Takeda_theorem" title="Sherman–Takeda theorem">Sherman–Takeda theorem</a></li> <li><a href="/wiki/Unbounded_operator" title="Unbounded operator">Unbounded operator</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Examples</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="/wiki/Wiener_algebra" title="Wiener algebra">Wiener algebra</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</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="/wiki/Almost_Mathieu_operator" title="Almost Mathieu operator">Almost Mathieu operator</a></li> <li><a href="/wiki/Corona_theorem" title="Corona theorem">Corona theorem</a></li> <li><a href="/wiki/Hearing_the_shape_of_a_drum" title="Hearing the shape of a drum">Hearing the shape of a drum</a> (<a href="/wiki/Dirichlet_eigenvalue" title="Dirichlet eigenvalue">Dirichlet eigenvalue</a>)</li> <li><a href="/wiki/Heat_kernel" title="Heat kernel">Heat kernel</a></li> <li><a href="/wiki/Kuznetsov_trace_formula" title="Kuznetsov trace formula">Kuznetsov trace formula</a></li> <li><a href="/wiki/Lax_pair" title="Lax pair">Lax pair</a></li> <li><a href="/wiki/Proto-value_function" title="Proto-value function">Proto-value function</a></li> <li><a href="/wiki/Ramanujan_graph" title="Ramanujan graph">Ramanujan graph</a></li> <li><a href="/wiki/Rayleigh%E2%80%93Faber%E2%80%93Krahn_inequality" title="Rayleigh–Faber–Krahn inequality">Rayleigh–Faber–Krahn inequality</a></li> <li><a href="/wiki/Spectral_geometry" title="Spectral geometry">Spectral geometry</a></li> <li><a class="mw-selflink selflink">Spectral method</a></li> <li><a href="/wiki/Spectral_theory_of_ordinary_differential_equations" title="Spectral theory of ordinary differential equations">Spectral theory of ordinary differential equations</a></li> <li><a href="/wiki/Sturm%E2%80%93Liouville_theory" title="Sturm–Liouville theory">Sturm–Liouville theory</a></li> <li><a href="/wiki/Superstrong_approximation" title="Superstrong approximation">Superstrong approximation</a></li> <li><a href="/wiki/Transfer_operator" title="Transfer operator">Transfer operator</a></li> <li><a href="/wiki/Transform_theory" title="Transform theory">Transform theory</a></li> <li><a href="/wiki/Weyl_law" title="Weyl law">Weyl law</a></li> <li><a href="/wiki/Wiener%E2%80%93Khinchin_theorem" title="Wiener–Khinchin theorem">Wiener–Khinchin theorem</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Spectral_method&oldid=1205120684">https://en.wikipedia.org/w/index.php?title=Spectral_method&oldid=1205120684</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Numerical_analysis" title="Category:Numerical analysis">Numerical analysis</a></li><li><a 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Spectral method - Wikipedia