Spectral method

Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain differential equations.Consequently, spectral methods connect variables globally while finite elements do so locally.Partially for this reason, spectral methods have excellent error properties, with the so-called "exponential convergence" being the fastest possible, when the solution is smooth.Spectral methods can be used to solve differential equations (PDEs, ODEs, eigenvalue, etc) and optimization problems.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 numerical method for ODEs.The implementation of the spectral method is normally accomplished either with collocation or a Galerkin or a Tau approach .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 Fourier transforms).For larger problems and nonsmooth solutions, finite elements will generally work better due to sparse matrices and better modelling of discontinuities and sharp bends.Here we presume an understanding of basic multivariate calculus and Fourier series.) then we are interested in finding a function f(x,y) so that where the expression on the left denotes the second partial derivatives of f in x and y, respectively.If we write f and g in Fourier series: and substitute into the differential equation, we obtain this equation: We have exchanged partial differentiation with an infinite sum, which is legitimate if we assume for instance that f has a continuous second derivative.With periodic boundary conditions, the Poisson equation possesses a solution only if b0,0 = 0.Since we're only interested in a finite window of frequencies (of size n, say) this can be done using a fast Fourier transform algorithm.We wish to solve the forced, transient, nonlinear Burgers' equation using a spectral approach.Integrating by parts and using periodicity grants To apply the Fourier–Galerkin method, choose both and where, this coupled system of ordinary differential equations may be integrated in time (using, e.g., a Runge Kutta technique) to find a solution.The nonlinear term is a convolution, and there are several transform-based techniques for evaluating it efficiently.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 aHowever, 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 elliptic boundary value problems.
applied mathematicsscientific computingdifferential equationsbasis functionsFourier seriessinusoidsfinite-element methodscompact supportsmoothshock capturingspectral-element methodoptimization problemsnumerical method for ODEsSteven OrszagcollocationGalerkinFourier transformscalculusPoisson equationfast Fourier transformBurgers' equationviscosityinner productIntegrating by partsGalerkin methodorthogonalityKronecker deltaRunge Kuttaconvolutionspectral element methodfinite element methodelliptic boundary value problemsGaussian gridPseudo-spectral methodCollocation methodHussaini M. Y.Numerical methods for partial differential equationsFinite differenceParabolicForward-time central-spaceCrank–NicolsonHyperbolicLax–FriedrichsLax–WendroffMacCormackUpwindMethod of characteristicsAlternating direction-implicitFinite-difference frequency-domainFinite-difference time-domainFinite volumeGodunovHigh-resolutionMonotonic upstream-centeredAdvection upstream-splittingRiemann solverEssentially non-oscillatoryWeighted essentially non-oscillatoryFinite elementhp-FEMExtendedDiscontinuous GalerkinSpectral elementMortarGradient discretisationLoubignac iterationSmoothedMeshless/MeshfreeSmoothed-particle hydrodynamicsPeridynamicsMoving particle semi-implicit methodMaterial point methodParticle-in-cellDomain decompositionSchur complementFictitious domainSchwarz alternatingadditiveabstract additiveNeumann–DirichletNeumann–NeumannPoincaré–Steklov operatorBalancingBalancing by constraintsTearing and interconnectFETI-DPPseudospectralMethod of linesMultigridLevel-setBoundary elementMethod of momentsImmersed boundaryAnalytic elementIsogeometric analysisInfinite difference methodInfinite element methodPetrov–Galerkin methodValidated numericsComputer-assisted proofIntegrable algorithmMethod of fundamental solutionsFunctional analysistopicsglossaryBanachFréchetHilbertHölderNuclearOrliczSchwartzSobolevTopological vectorBarrelledCompleteLocally convexReflexiveSeparableHahn–BanachRiesz representationClosed graphUniform boundedness principleKrein–MilmanMin–maxGelfand–NaimarkBanach–AlaogluAdjointBoundedCompactHilbert–SchmidtNormalTrace classTransposeUnboundedUnitaryBanach algebraC*-algebraSpectrum of a C*-algebraOperator algebraGroup algebra of a locally compact groupVon Neumann algebraInvariant subspace problemMahler's conjectureHardy spaceSpectral theory of ordinary differential equationsHeat kernelIndex theoremCalculus of variationsFunctional calculusIntegral linear operatorJones polynomialTopological quantum field theoryNoncommutative geometryRiemann hypothesisDistributionGeneralized functionsApproximation propertyBalanced setChoquet theoryWeak topologyBanach–Mazur distanceTomita–Takesaki theorySpectral theory*-algebrasInvolution/*-algebraB*-algebraNoncommutative topologyProjection-valued measureSpectrumSpectral radiusOperator spaceGelfand–Mazur theoremGelfand–Naimark theoremGelfand representationPolar decompositionSingular value decompositionSpectral theoremSpectral theory of normal C*-algebrasIsospectraloperatorHermitian/Self-adjointKrein–Rutman theoremNormal eigenvalueSpectral asymmetrySpectral gapDecomposition of a spectrumContinuousDirect integralDiscreteSpectral abscissaBorel functional calculusMin-max theoremPositive operator-valued measureRiesz projectorRigged Hilbert spaceSpectral theory of compact operatorsAmenable Banach algebraApproximate identityBanach function algebraDisk algebraNuclear C*-algebraUniform algebraAlon–Boppana boundBauer–Fike theoremNumerical rangeSchur–Horn theoremDirac spectrumEssential spectrumPseudospectrumStructure spaceShilov boundaryAbstract index groupBanach algebra cohomologyCohen–Hewitt factorization theoremExtensions of symmetric operatorsFredholm theoryLimiting absorption principleSchröder–Bernstein theorems for operator algebrasSherman–Takeda theoremUnbounded operatorWiener algebraAlmost Mathieu operatorCorona theoremHearing the shape of a drumDirichlet eigenvalueKuznetsov trace formulaLax pairProto-value functionRamanujan graphRayleigh–Faber–Krahn inequalitySpectral geometrySturm–Liouville theorySuperstrong approximationTransfer operatorTransform theoryWeyl lawWiener–Khinchin theorem