Chebyshev pseudospectral method and waveform relaxation for differential and differential-functional parabolic equations
type: Item
instance of: scholarly article
title: Chebyshev pseudospectral method and waveform relaxation for differential and differential-functional parabolic equations
zbMATH Open document ID: 0948.65102
DOI: 10.1016/S0168-9274(99)00135-X
author: B. Zubik-Kowal
published in: Applied Numerical Mathematics
publication date: 2000-11-22T00:00:00Z
review text: For the one-dimensional heat equation with or without a linear functional term (covering e.g. linear integro-differential equations or equations with a time delay), the numerical solution by means of the waveform relaxation is studied. After the spatial discretization, which uses finite differences of second-order or a pseudospectral method based upon Chebyshev polynomials of the first kind, the Gauß-Seidel waveform relaxation technique is applied. This results in an iterative sequence of systems of ordinary differential equations, each described by a lower triangle matrix. Hence, a solution of the equations from up to down by e.g. the backward differentiation formulae (BDF), as proposed by the author, does not require the solution of an algebraic system. For both, the waveform relaxation after the finite difference method and after the Chebyshev method, resp., the time-dependent error is analyzed by means of estimates using logarithmic norms. Numerical examples illustrate the two different methods. Here, for the time discretization, the BDF of order 3 with constant step size has been used. Better convergence and computational efficiency has been observed when using the Chebyshev method instead of the method of finite differences.
Mathematics Subject Classification ID: 65M70
Mathematics Subject Classification ID: 65M15
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 35R10
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 35K15
zbMATH DE Number: 1462306
zbMATH Keywords: Chebyshev pseudospectral method
zbMATH Keywords: waveform relaxation
zbMATH Keywords: differential-functional parabolic equations
zbMATH Keywords: error bounds
zbMATH Keywords: numerical examples
zbMATH Keywords: heat equation
zbMATH Keywords: linear integro-differential equations
zbMATH Keywords: backward differentiation formulae
zbMATH Keywords: finite difference method
zbMATH Keywords: convergence
zbMATH Keywords: computational efficiency
reviewed by: Etienne Emmrich
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1462306