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September 2006 Fixed-point Iterative Sweeping Methods for Static Hamilton-Jacobi Equations
Yong-Tao Zhang, Hong-Kai Zhao, Shanqin Chen
Methods Appl. Anal. 13(3): 299-320 (September 2006).


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.


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Yong-Tao Zhang. Hong-Kai Zhao. Shanqin Chen. "Fixed-point Iterative Sweeping Methods for Static Hamilton-Jacobi Equations." Methods Appl. Anal. 13 (3) 299 - 320, September 2006.


Published: September 2006
First available in Project Euclid: 18 January 2008

zbMATH: 1134.65076
MathSciNet: MR2382437

Primary: 65N99
Secondary: 35L60

Keywords: Eikonal equations , fast sweeping methods , Gauss-Seidel iteration , Jacobi iteration , static Hamilton-Jacobi equations

Rights: Copyright © 2006 International Press of Boston

Vol.13 • No. 3 • September 2006
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