Spectral Methods for Initial Boundary Value Problems--An Alternative Approach

This paper presents a new approach to spectral methods for initial boundary value problems. A filtered version of the partial differential equation and the initial and boundary conditions at an overdetermined set of points are collocated. As an approximate solution, the function is chosen that belon...

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Bibliographic Details
Published in:SIAM journal on numerical analysis Vol. 27; no. 4; pp. 885 - 903
Main Author: Dutt, P.
Format: Journal Article
Language:English
Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.08.1990
Subjects:
Accuracy
Applied mathematics
Boundary conditions
Boundary value problems
Continuous functions
Degrees of polynomials
Eigenvalues
Exact sciences and technology
Mathematics
Methods
Numerical analysis
Numerical analysis. Scientific computation
Parameter estimation
Partial differential equations
Partial differential equations, initial value problems and time-dependant initial-boundary value problems
Polynomials
Preliminary estimates
Sciences and techniques of general use
Spectral methods
Vector spaces
Quadrature
Boundary value problem
Multigrid
Tchebychev polynomial
Initial value problem
Partial differential equation
ISSN:0036-1429, 1095-7170
Online Access:Get full text
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Summary:This paper presents a new approach to spectral methods for initial boundary value problems. A filtered version of the partial differential equation and the initial and boundary conditions at an overdetermined set of points are collocated. As an approximate solution, the function is chosen that belongs to an appropriate finite-dimensional space and minimizes a weighted average of the residuals at these points. It is proved that the approximate solution converges to the actual solution at a spectral rate of accuracy in both space and time. The proof is based on a priori energy estimates that have been proved for such systems. Although this method is restricted here to hyperbolic initial boundary value problems and Chebyshev polynomials, it generalizes to general initial boundary value problems, boundary value problems, and Gegenbauer polynomials.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
content type line 14
ISSN:0036-1429
1095-7170
DOI:10.1137/0727051