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Volume 11, Issue 1
Fixed-Point Fast Sweeping WENO Methods for Steady State Solution of Scalar Hyperbolic Conservation Law

S. Chen

Int. J. Numer. Anal. Mod., 11 (2014), pp. 117-130.

Published online: 2014-11

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  • Abstract

Fast sweeping methods were developed in the literature to efficiently solve static Hamilton-Jacobi equations. This class of methods utilize the Gauss-Seidel iterations and alternating sweeping strategy to achieve fast convergence rate. They take advantage of the properties of hyperbolic partial differential equations (PDEs) and try to cover a family of characteristics of the corresponding Hamilton-Jacobi equation in a certain direction simultaneously in each sweeping order. In [16], the Gauss-Seidel idea and alternating sweeping strategy were adopted to the time-marching type fixed-point iterations to solve the static Hamilton-Jacobi equations, and numerical examples verified at least a 2 times acceleration of convergence even on relatively coarse grids. In this paper, we apply the same approach to solve steady state solution of hyperbolic conservation laws. We use numerical examples to verify that a 2 times acceleration of convergence is achieved. And the computational cost is exactly the same as the time-marching scheme at each iteration. Based on the Gauss-Seidel iterations, we explore the successive overrelaxation (SOR) approach to further improve the performance of our fixed-point sweeping methods.

  • Keywords

fast sweeping methods, WENO methods, Jacobi iteration, Gauss-Seidel iteration, hyperbolic conservation laws, steady state.

  • AMS Subject Headings

35R35, 49J40, 60G40

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-11-117, author = {}, title = {Fixed-Point Fast Sweeping WENO Methods for Steady State Solution of Scalar Hyperbolic Conservation Law }, journal = {International Journal of Numerical Analysis and Modeling}, year = {2014}, volume = {11}, number = {1}, pages = {117--130}, abstract = {

Fast sweeping methods were developed in the literature to efficiently solve static Hamilton-Jacobi equations. This class of methods utilize the Gauss-Seidel iterations and alternating sweeping strategy to achieve fast convergence rate. They take advantage of the properties of hyperbolic partial differential equations (PDEs) and try to cover a family of characteristics of the corresponding Hamilton-Jacobi equation in a certain direction simultaneously in each sweeping order. In [16], the Gauss-Seidel idea and alternating sweeping strategy were adopted to the time-marching type fixed-point iterations to solve the static Hamilton-Jacobi equations, and numerical examples verified at least a 2 times acceleration of convergence even on relatively coarse grids. In this paper, we apply the same approach to solve steady state solution of hyperbolic conservation laws. We use numerical examples to verify that a 2 times acceleration of convergence is achieved. And the computational cost is exactly the same as the time-marching scheme at each iteration. Based on the Gauss-Seidel iterations, we explore the successive overrelaxation (SOR) approach to further improve the performance of our fixed-point sweeping methods.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/517.html} }
TY - JOUR T1 - Fixed-Point Fast Sweeping WENO Methods for Steady State Solution of Scalar Hyperbolic Conservation Law JO - International Journal of Numerical Analysis and Modeling VL - 1 SP - 117 EP - 130 PY - 2014 DA - 2014/11 SN - 11 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/517.html KW - fast sweeping methods, WENO methods, Jacobi iteration, Gauss-Seidel iteration, hyperbolic conservation laws, steady state. AB -

Fast sweeping methods were developed in the literature to efficiently solve static Hamilton-Jacobi equations. This class of methods utilize the Gauss-Seidel iterations and alternating sweeping strategy to achieve fast convergence rate. They take advantage of the properties of hyperbolic partial differential equations (PDEs) and try to cover a family of characteristics of the corresponding Hamilton-Jacobi equation in a certain direction simultaneously in each sweeping order. In [16], the Gauss-Seidel idea and alternating sweeping strategy were adopted to the time-marching type fixed-point iterations to solve the static Hamilton-Jacobi equations, and numerical examples verified at least a 2 times acceleration of convergence even on relatively coarse grids. In this paper, we apply the same approach to solve steady state solution of hyperbolic conservation laws. We use numerical examples to verify that a 2 times acceleration of convergence is achieved. And the computational cost is exactly the same as the time-marching scheme at each iteration. Based on the Gauss-Seidel iterations, we explore the successive overrelaxation (SOR) approach to further improve the performance of our fixed-point sweeping methods.

S. Chen. (1970). Fixed-Point Fast Sweeping WENO Methods for Steady State Solution of Scalar Hyperbolic Conservation Law . International Journal of Numerical Analysis and Modeling. 11 (1). 117-130. doi:
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