High order fixed-point sweeping WENO methods for steady state of hyperbolic conservation laws and its convergence study

Liang Wu University of Notre Dame Yong-Tao Zhang University of Notre Dame Shuhai Zhang China Aerodynamics Research and Development Center Chi-Wang Shu Brown University

Numerical Analysis and Scientific Computing mathscidoc:1610.25015

Communications in Computational Physics, 20, 835-869, 2016
Fixed-point iterative sweeping methods were developed in the literature to efficiently solve static Hamilton-Jacobi equations. This class of methods utilizes 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. Different from other fast sweeping methods, fixed-point iterative sweeping methods have the advantages such as that they have explicit forms and do {\it not} involve inverse operation of nonlinear local systems. In principle, it can be applied in solving very general equations using any monotone numerical fluxes and high order approximations easily. In this paper, based on the recently developed fifth order WENO schemes which improve the convergence of the classical WENO schemes by removing slight post-shock oscillations, we design fifth order fixed-point sweeping WENO methods for efficient computation of steady state solution of hyperbolic conservation laws. Especially, we show that although the methods do {\it not} have linear computational complexity, they converge to steady state solutions much faster than regular time-marching approach by stability improvement for high order schemes with a forward Euler time-marching.
Fixed-point sweeping methods, WENO methods, high order accuracy, steady state, hyperbolic conservation laws, convergence
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@inproceedings{liang2016high,
  title={High order fixed-point sweeping WENO methods for steady state of hyperbolic conservation laws and its convergence study},
  author={Liang Wu, Yong-Tao Zhang, Shuhai Zhang, and Chi-Wang Shu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161011104425713079129},
  booktitle={Communications in Computational Physics},
  volume={20},
  pages={835-869},
  year={2016},
}
Liang Wu, Yong-Tao Zhang, Shuhai Zhang, and Chi-Wang Shu. High order fixed-point sweeping WENO methods for steady state of hyperbolic conservation laws and its convergence study. 2016. Vol. 20. In Communications in Computational Physics. pp.835-869. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161011104425713079129.
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