Spectral method: Difference between revisions

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{{More footnotes|date=August 2013}}

'''Spectral methods''' are a class of techniques used in [[applied mathematics]] and [[scientific computing]] to numerically solve certain [[differential equation]]s, potentially involving the use of the [[fast Fourier transform]]. The idea is to write the solution of the differential equation as a sum of certain "[[basis function]]s" (for example, as a [[Fourier series]] which is a sum of [[Sine wave|sinusoid]]s) and then to choose the coefficients in the sum in order to satisfy the differential equation as well as possible.

Spectral methods and [[finite element method]]s 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. In([[compact othersupport]]). wordsConsequently, spectral methods takeconnect on avariables ''global approachglobally'' while finite elementelements methodsdo use aso ''local approachlocally''. 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 function|smooth]]. However, there are no known three-dimensional single ___domain spectral [[shock capturing]] results (shock waves are not smooth).<ref name="CHQZ">[https://books.google.com/books?id=7COgEw5_EBQC pp 235, Spectral Methods]: evolution to complex geometries and applications to fluid dynamics, By Canuto, Hussaini, Quarteroni and Zang, Springer, 2007.</ref> In the finite element community, a method where the degree of the elements is very high or increases as the grid parameter ''h'' decreases to zero is sometimes called a [[spectral element method]].

Spectral methods can be used to solve [[ordinary differential equations]] (PDEs, ODEs), [[partialeigenvalue, differential equations]] (PDEsetc) and [[eigenvalueoptimization problem]] problems involving differential equationss. 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 methods for ordinary differential equations|numerical method for ODEs]]. Eigenvalue problems for ODEs are similarly converted to matrix eigenvalue problems {{Citation needed|date=August 2013}}.

Spectral methods were developed in a long series of papers by [[Steven Orszag]] 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 [[collocation method|collocation]] or a [[Galerkin method|Galerkin]] or a [[Tau method|Tau]] approach . For very small problems, the spectral method is unique that solutions may be written out symbolically, yielding a practical alternative to series solutions for differential equations.

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 transform]]s). For larger problems and nonsmooth solutions, finite elements will generally work better due to sparse matrices and better modelling of discontinuities and sharp bends.

Spectral methods are computationally less expensive than finite element methods, but become less accurate for problems with complex geometries and discontinuous coefficients. This increase in error is a consequence of the [[Gibbs phenomenon]].

==Examples of spectral methods==