Lax–Friedrichs Multigrid Fast Sweeping Methods for Steady State Problems for Hyperbolic Conservation Laws

Abstract

Fast sweeping methods are efficient Gauss---Seidel iterative numerical schemes originally designed for solving static Hamilton---Jacobi equations. Recently, these methods have been applied to solve hyperbolic conservation laws with source terms. In this paper, we propose Lax---Friedrichs fast sweeping multigrid methods which allow even more efficient calculations of viscosity solutions of stationary hyperbolic problems. Due to the choice of Lax---Friedrichs numerical fluxes, general problems can be solved without difficult inversion. High order discretization, e.g., WENO finite difference method, can be incorporated to achieve high order accuracy. On the other hand, multigrid methods, which have been widely used to solve elliptic equations, can speed up the computation by smoothing errors of low frequencies on coarse meshes. We modify the classical multigrid method with regard to properties of viscous solutions to hyperbolic conservation equations by introducing WENO interpolation between levels of mesh grids. Extensive numerical examples in both scalar and system test problems in one and two dimensions demonstrate the efficiency, high order accuracy and the capability of resolving singularities of the viscosity solutions.