Communications on Applied Mathematics and Computation >

2023 , Vol. 5 >Issue 1: 403 - 427

DOI: https://doi.org/10.1007/s42967-021-00179-6

ORIGINAL PAPERS

A Fixed-Point Fast Sweeping WENO Method with Inverse Lax-Wendroff Boundary Treatment for Steady State of Hyperbolic Conservation Laws

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  • 1 Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles(NUAA), MIIT, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, Jiangsu, China;
    2 State Key Laboratory of Mechanics and Control of Mechanical Structures and Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles(NUAA), MIIT, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, Jiangsu, China;
    3 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA;
    4 Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN 46556, USA

Received date: 2021-07-25

  Revised date: 2021-11-14

  Online published: 2023-03-08

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Abstract

Fixed-point fast sweeping WENO methods are a class of efficient high-order numerical methods to solve steady-state solutions of hyperbolic partial differential equations (PDEs). The Gauss-Seidel iterations and alternating sweeping strategy are used to cover characteristics of hyperbolic PDEs in each sweeping order to achieve fast convergence rate to steady-state solutions. A nice property of fixed-point fast sweeping WENO methods which distinguishes them from other fast sweeping methods is that they are explicit and do not require inverse operation of nonlinear local systems. Hence, they are easy to be applied to a general hyperbolic system. To deal with the difficulties associated with numerical boundary treatment when high-order finite difference methods on a Cartesian mesh are used to solve hyperbolic PDEs on complex domains, inverse Lax-Wendroff (ILW) procedures were developed as a very effective approach in the literature. In this paper, we combine a fifthorder fixed-point fast sweeping WENO method with an ILW procedure to solve steadystate solution of hyperbolic conservation laws on complex computing regions. Numerical experiments are performed to test the method in solving various problems including the cases with the physical boundary not aligned with the grids. Numerical results show highorder accuracy and good performance of the method. Furthermore, the method is compared with the popular third-order total variation diminishing Runge-Kutta (TVD-RK3) time-marching method for steady-state computations. Numerical examples show that for most of examples, the fixed-point fast sweeping method saves more than half CPU time costs than TVD-RK3 to converge to steady-state solutions.

Key words： Fixed-point fast sweeping methods; Multi-resolution WENO schemes; Steady state; ILW procedure; Convergence

Cite this article

Liang Li, Jun Zhu, Chi-Wang Shu, Yong-Tao Zhang . A Fixed-Point Fast Sweeping WENO Method with Inverse Lax-Wendroff Boundary Treatment for Steady State of Hyperbolic Conservation Laws[J]. Communications on Applied Mathematics and Computation, 2023 , 5(1) : 403 -427 . DOI: 10.1007/s42967-021-00179-6

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