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Collocation Methods for Hyperbolic Partial Differential Equations with Singular Sources

Collocation Methods for Hyperbolic Partial Differential Equations with Singular Sources
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Year:    2009

Author:    Jae-Hun Jung, Wai Sun Don

Advances in Applied Mathematics and Mechanics, Vol. 1 (2009), Iss. 6 : pp. 769–780

Abstract

A numerical study is given on the spectral methods and the high order WENO finite difference scheme for the solution of linear and nonlinear hyperbolic partial differential equations with stationary and non-stationary singular sources. The singular source term is represented by the $δ$-function. For the approximation of the $δ$-function, the direct projection method is used that was proposed in [6]. The $δ$-function is constructed in a consistent way to the derivative operator. Nonlinear sine-Gordon equation with a stationary singular source was solved with the Chebyshev collocation method. The $δ$-function with the spectral method is highly oscillatory but yields good results with small number of collocation points. The results are compared with those computed by the second order finite difference method. In modeling general hyperbolic equations with a non-stationary singular source, however, the solution of the linear scalar wave equation with the non-stationary singular source using the direct projection method yields non-physical oscillations for both the spectral method and the WENO scheme. The numerical artifacts arising when the non-stationary singular source term is considered on the discrete grids are explained.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/aamm.09-m09S10

Advances in Applied Mathematics and Mechanics, Vol. 1 (2009), Iss. 6 : pp. 769–780

Published online:    2009-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    12

Keywords:    Singular sources Dirac-$\delta$-function Direct projection method Chebyshev collocation method WENO scheme.

Author Details

Jae-Hun Jung

Wai Sun Don

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