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Volume 5, Issue 3
The Dissipative Spectral Methods for the First Order Linear Hyperbolic Equations

Lian Chen, Zhongqiang Zhang & Heping Ma

DOI: 10.4208/nmtma.2012.m1110

Numer. Math. Theor. Meth. Appl., 5 (2012), pp. 493-508.

Published online: 2012-05

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  • Abstract

In this paper, we introduce the dissipative spectral methods (DSM) for the first order linear hyperbolic equations in one dimension. Specifically, we consider the Fourier DSM for periodic problems and the Legendre DSM for equations with the Dirichlet boundary condition. The error estimates of the methods are shown to be quasi-optimal for variable-coefficients equations. Numerical results are given to verify high accuracy of the DSM and to compare the proposed schemes with some high performance methods, showing some superiority in long-term integration for the periodic case and in dealing with limited smoothness near or at the boundary for the Dirichlet case.

  • Keywords

First order hyperbolic equation, dissipative spectral method, error estimate.

  • AMS Subject Headings

65N15, 65N35

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-5-493, author = {Lian Chen, Zhongqiang Zhang and Heping Ma}, title = {The Dissipative Spectral Methods for the First Order Linear Hyperbolic Equations}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2012}, volume = {5}, number = {3}, pages = {493--508}, abstract = {

In this paper, we introduce the dissipative spectral methods (DSM) for the first order linear hyperbolic equations in one dimension. Specifically, we consider the Fourier DSM for periodic problems and the Legendre DSM for equations with the Dirichlet boundary condition. The error estimates of the methods are shown to be quasi-optimal for variable-coefficients equations. Numerical results are given to verify high accuracy of the DSM and to compare the proposed schemes with some high performance methods, showing some superiority in long-term integration for the periodic case and in dealing with limited smoothness near or at the boundary for the Dirichlet case.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2012.m1110}, url = {http://global-sci.org/intro/article_detail/nmtma/5947.html} }
TY - JOUR T1 - The Dissipative Spectral Methods for the First Order Linear Hyperbolic Equations AU - Lian Chen, Zhongqiang Zhang & Heping Ma JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 493 EP - 508 PY - 2012 DA - 2012/05 SN - 5 DO - http://doi.org/10.4208/nmtma.2012.m1110 UR - https://global-sci.org/intro/article_detail/nmtma/5947.html KW - First order hyperbolic equation, dissipative spectral method, error estimate. AB -

In this paper, we introduce the dissipative spectral methods (DSM) for the first order linear hyperbolic equations in one dimension. Specifically, we consider the Fourier DSM for periodic problems and the Legendre DSM for equations with the Dirichlet boundary condition. The error estimates of the methods are shown to be quasi-optimal for variable-coefficients equations. Numerical results are given to verify high accuracy of the DSM and to compare the proposed schemes with some high performance methods, showing some superiority in long-term integration for the periodic case and in dealing with limited smoothness near or at the boundary for the Dirichlet case.

Lian Chen, Zhongqiang Zhang and Heping Ma. (2012). The Dissipative Spectral Methods for the First Order Linear Hyperbolic Equations. Numerical Mathematics: Theory, Methods and Applications. 5 (3). 493-508. doi:10.4208/nmtma.2012.m1110
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