Fixed-point iterative sweeping methods for static Hamilton-Jacobi equations

Fast sweeping methods utilize the Gauss-Seidel iterations and alternating sweeping strat-egy to achieve the fast convergence for computations of static Hamilton-Jacobi equations. They take advantage of the properties of hyperbolic PDEs and try to cover a family of characteristics of the corresponding Hamilton-Jacobi equation in a certain direction simul-taneously in each sweeping order. The time-marching approach to steady state calculation is much slower than the fast sweeping methods due to the CFL condition constraint. But this kind of fixed-point iterations as time-marching methods have explicit form and do not involve inverse operation of nonlinear Hamiltonian. So it can solve general Hamilton-Jacobi equations using any monotone numerical Hamiltonian and high order approximations eas-ily. In this paper, we adopt the Gauss-Seidel idea and alternating sweeping strategy to the time-marching type fixed-point iterations to solve the static Hamilton-Jacobi equations. Ex-tensive numerical examples verify at least a 2 ∼ 5 times acceleration of convergence even on relatively coarse grids. The acceleration is even more when the grid is further refined. More-over the Gauss-Seidel philosophy and alternating sweeping strategy improves the stability, i.e., a larger CFL number can be used. Also the computational cost is exactly the same as the time-marching scheme at each time step