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High Order Fixed-Point Sweeping WENO Methods for Steady State of Hyperbolic Conservation Laws and Its Convergence Study

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Partial differential equations, boundary value problems

Published online by Cambridge University Press:  05 October 2016

Liang Wu ,
Yong-Tao Zhang ,
Shuhai Zhang  and
Chi-Wang Shu
Liang Wu*
Affiliation:
Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN 46556, USA
Yong-Tao Zhang*
Affiliation:
Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN 46556, USA
Shuhai Zhang*
Affiliation:
State Key Laboratory of Aerodynamics, China Aerodynamics Research and Development Center, Mianyang, Sichuan 621000, China
Chi-Wang Shu*
Affiliation:
Division of Applied Mathematics, Brown University, Providence, RI 02912, USA
*
*Corresponding author. Email addresses:lwu2@alumni.nd.edu (L. Wu), yzhang10@nd.edu (Y.-T. Zhang), shuhai_zhang@163.com (S. Zhang), shu@dam.brown.edu (C.-W. Shu)
*Corresponding author. Email addresses:lwu2@alumni.nd.edu (L. Wu), yzhang10@nd.edu (Y.-T. Zhang), shuhai_zhang@163.com (S. Zhang), shu@dam.brown.edu (C.-W. Shu)
*Corresponding author. Email addresses:lwu2@alumni.nd.edu (L. Wu), yzhang10@nd.edu (Y.-T. Zhang), shuhai_zhang@163.com (S. Zhang), shu@dam.brown.edu (C.-W. Shu)
*Corresponding author. Email addresses:lwu2@alumni.nd.edu (L. Wu), yzhang10@nd.edu (Y.-T. Zhang), shuhai_zhang@163.com (S. Zhang), shu@dam.brown.edu (C.-W. Shu)
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Abstract

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 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 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.

Keywords

Fixed-point sweeping methodsWENO methodshigh order accuracysteady statehyperbolic conservation lawsconvergence

MSC classification

Secondary: 65M06: Finite difference methods 65M12: Stability and convergence of numerical methods 65N06: Finite difference methods 65N12: Stability and convergence of numerical methods
Type
Research Article
Information
Communications in Computational Physics , Volume 20 , Issue 4 , October 2016 , pp. 835 - 869
Copyright
Copyright © Global-Science Press 2016 

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