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 talk, we will review fast sweeping methods on Hamilton-Jacobi equation and generalize the fast sweeping methods to hyperbolic conservation laws with source terms. Among fast sweeping schemes for Hamilton-Jacobi equations, the methods based on upwind Hamiltonians are most efficient for convex Hamiltonians, while the methods based on Lax-Friedrichs Hamiltonians are most flexible to deal with general non-convex Hamiltonians. We thus extend the method based on Lax-Friedrichs fluxes to solve steady state problems for hyperbolic conservation laws. 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. 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.
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