Fast sweeping methods utilize the Gauss-Seidel iterations and alternating
sweeping strategy 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 simultaneously
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
easily. 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. Extensive numerical examples verify at
least a $2\sim5$ times acceleration of convergence even on relatively coarse
grids. The acceleration is even more when the grid is further refined.
Moreover 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.