Spectral method: Difference between revisions

Revision as of 22:42, 8 February 2024

Revision as of 22:43, 8 February 2024

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'''Spectral methods''' are a class of techniques used in [[applied mathematics]] and [[scientific computing]] to numerically solve certain [[differential equation]]s. 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 ([[compact support]]). 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 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'' increases is sometimes called a [[spectral element method]].

Spectral methods can be used to solve [[differential equations]] (PDEs, ODEs, eigenvalue, etc) and [[optimization problem]]s. 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}}.