Methods and Applications of Analysis

Volume 13 (2006)

Number 3

Fixed-point Iterative Sweeping Methods for Static Hamilton-Jacobi Equations

Pages: 299 – 320

DOI: http://dx.doi.org/10.4310/MAA.2006.v13.n3.a5

Authors

Shanqin Chen

Yong-Tao Zhang

Hong-Kai Zhao

Abstract

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.

Keywords

fast sweeping methods; Jacobi iteration; Gauss-Seidel iteration; static Hamilton-Jacobi equations; Eikonal equations

2010 Mathematics Subject Classification

Primary 65N99. Secondary 35L60.

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