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

TL;DR

This paper introduces an absolutely convergent fixed-point fast sweeping WENO method for steady hyperbolic conservation laws, overcoming previous convergence issues and ensuring residuals reach machine precision.

Contribution

It develops a new fixed-point sweeping WENO scheme with guaranteed convergence by integrating multi-resolution WENO techniques.

Findings

Residue converges to machine zero for all benchmark problems.

Faster convergence compared to traditional TVD Runge-Kutta methods.

Applicable to general hyperbolic equations with high order accuracy.

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. Furthermore, good properties of fixed-point iterative 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 [L. Wu, Y.-T. Zhang, S. Zhang and C.-W. Shu, Commun. Comput. Phys., 20 (2016)], a fifth order fixed-point sweeping WENO scheme was designed and it was shown that the scheme converges much faster than the total variation diminishing (TVD) Runge-Kutta 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 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 [J. Zhu and C.-W. Shu, J. Comput. Phys., 375 (2018)] 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.