Spectral method: Difference between revisions

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Spectral methods can be used to solve [[ordinary differential equations]] (ODEs), [[partial differential equations]] (PDEs) and [[eigenvalue]] problems involving differential equations. 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. Later, the possible usage of [[compressive sensing]] for the enhancement of the spectral methods is also proposed and discussed. <ref>[https://www.nature.com/articles/srep22100 Compressive spectral method for the simulation of nonlinear gravity waves], Cihan Bayindir, ''Scientific Reports'' 6, 22100, 25 February 2016. </ref> 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.

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]].