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

• Published in: Communications on Applied Mathematics and Computation (Online) Vol. 5; no. 1; pp. 403 - 427
• Main Authors: Li, Liang, Zhu, Jun, Shu, Chi-Wang, Zhang, Yong-Tao
• Format: Journal Article
• Language: English
• Published: Singapore Springer Nature Singapore 01.03.2023
• Subjects: Applications of Mathematics; Computational Science and Engineering; Mathematics; Mathematics and Statistics; Original Paper; 65M06; 65N06; Multi-resolution WENO schemes; Fixed-point fast sweeping methods; Steady state; 65N22; ILW procedure; 35L65; Convergence
• ISSN: 2096-6385; 2661-8893
• DOI: 10.1007/s42967-021-00179-6

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 fifth-order fixed-point fast sweeping WENO method with an ILW procedure to solve steady-state 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 high-order 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.