A hybrid Fourier-Chebyshev method for partial differential equations

Rodrigo Platte, Anne Gelb

Research output: Contribution to journalArticle

12 Citations (Scopus)

Abstract

We propose a pseudospectral hybrid algorithm to approximate the solution of partial differential equations (PDEs) with non-periodic boundary conditions. Most of the approximations are computed using Fourier expansions that can be efficiently obtained by fast Fourier transforms. To avoid the Gibbs phenomenon, super-Gaussian window functions are used in physical space. Near the boundaries, we use local polynomial approximations to correct the solution. We analyze the accuracy and eigenvalue stability of the method for several PDEs. The method compares favorably to traditional spectral methods, and numerical results indicate that for hyperbolic problems a time step restriction of O(1/N) is sufficient for stability.

Original languageEnglish (US)
Pages (from-to)244-264
Number of pages21
JournalJournal of Scientific Computing
Volume39
Issue number2
DOIs
StatePublished - May 2009

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Chebyshev's Method
Fourier Method
Partial differential equations
Partial differential equation
Gibbs Phenomenon
Polynomial approximation
Local Polynomial
Hyperbolic Problems
Local Approximation
Fourier Expansion
Polynomial Approximation
Fast Fourier transform
Spectral Methods
Hybrid Algorithm
Fast Fourier transforms
Boundary conditions
Sufficient
Restriction
Eigenvalue
Numerical Results

Keywords

  • Exponential convergence
  • Fourier spectral method
  • Hybrid methods
  • Non-periodic boundary conditions
  • Time-dependent pdes

ASJC Scopus subject areas

  • Software
  • Computational Theory and Mathematics
  • Theoretical Computer Science
  • Engineering(all)

Cite this

A hybrid Fourier-Chebyshev method for partial differential equations. / Platte, Rodrigo; Gelb, Anne.

In: Journal of Scientific Computing, Vol. 39, No. 2, 05.2009, p. 244-264.

Research output: Contribution to journalArticle

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