Absolutely convergent fixed-point fast sweeping WENO methods for steady state of hyperbolic conservation laws

Abstract

Fixed-point iterative sweeping methods were developed in the literature to efficiently solve steady state solutions of Hamilton-Jacobi equations and hyperbolic conservation laws. Similar as other fast sweeping schemes, the key components of this class of methods are the Gauss-Seidel iterations and alternating sweeping strategy to achieve fast convergence rate. A family of characteristics of the corresponding hyperbolic partial differential equations (PDEs) are covered in a certain direction simultaneously in each sweeping order. Furthermore, good properties of fixed-point iterative sweeping methods which are different from other fast sweeping methods include that they have explicit forms and do not involve inverse operation of nonlinear local systems, and they can be applied to general hyperbolic equations using any monotone numerical fluxes and high order approximations easily. In a recent article (Wu et al. 2016) [31], a fifth order fixed-point sweeping WENO scheme was designed for solving steady state of hyperbolic conservation laws, and it was shown that the scheme converges to steady state solution much faster than the regular total variation diminishing (TVD) Runge-Kutta time-marching approach by stability improvement of high order schemes with a forward Euler time-marching. An open problem is that for some benchmark numerical examples, the iteration residue of the fixed-point sweeping WENO scheme hangs at a truncation error level, or even higher error levels, instead of settling down to machine zero. This issue makes it difficult to determine the convergence criterion for the iteration and challenging to apply the method to complex problems. To solve this issue, in this paper we apply the multi-resolution WENO scheme developed in Zhu and Shu (2018) [45] to the fifth order fixed-point sweeping WENO scheme and obtain an absolutely convergent fixed-point fast sweeping method for steady state of hyperbolic conservation laws, i.e., the residue of the fast sweeping iterations converges to machine zero / round off errors for all benchmark problems tested. Extensive numerical experiments, including solving difficult problems such as the shock reflection, supersonic flow past plates, and supersonic, transonic and subsonic flows past an airfoil, etc., are performed to show the accuracy, computational efficiency, and absolute convergence of the presented fifth order sweeping scheme.