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Fixed-point iterative sweeping methods for static Hamilton-Jacobi equations. (English) Zbl 1134.65076

Summary: 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 partial differential equations 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 Courant-Friedrichs-Levi (CFL) condition constraint. But this kind of fixed-point iterations as time-marching methods have explicit form and do not involve the inverse operation of the 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.

MSC:

65N06 Finite difference methods for boundary value problems involving PDEs
35F30 Boundary value problems for nonlinear first-order PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
65K10 Numerical optimization and variational techniques
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