Lax–Friedrichs fast sweeping methods for steady state problems for hyperbolic conservation laws

Fast sweeping methods are efficient iterative numerical schemes originally designed for solving stationary Hamilton–Jacobi equations. Their efficiency relies on Gauss–Seidel type nonlinear iterations, and a finite number of sweeping directions. In this paper, we generalize the fast sweeping methods to hyperbolic conservation laws with source terms. The algorithm is obtained through finite difference discretization, with the numerical fluxes evaluated in WENO (Weighted Essentially Non-oscillatory) fashion, coupled with Gauss–Seidel iterations. In particular, we consider mainly the Lax-Friedrichs numerical fluxes. 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 shocks of the proposed methods.