Fixed-point Fast Sweeping Weno Methods for Steady State Solution of Scalar Hyperbolic Conservation Laws

Fast sweeping methods were developed in the literature to efficiently solve static Hamilton-Jacobi equations. This class of methods utilize 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. In [16], the Gauss-Seidel idea and alternating sweeping strategy were adopted to the time-marching type fixed-point iterations to solve the static Hamilton-Jacobi equations, and numerical examples verified at least a 2 times acceleration of convergence even on relatively coarse grids. In this paper, we apply the same approach to solve steady state solution of hyperbolic conservation laws. We use numerical examples to verify that a 2 times acceleration of convergence is achieved. And the computational cost is exactly the same as the time-marching scheme at each iteration. Based on the Gauss-Seidel iterations, we explore the successive overrelaxation (SOR) approach to further improve the performance of our fixed-point sweeping methods.