High Order Fast Sweeping Methods for Static Hamilton–Jacobi Equations
We construct high order fast sweeping numerical methods for computing viscosity solutions of static Hamilton–Jacobi equations on rectangular grids. These methods combine high order weighted essentially non-oscillatory (WENO) approximations to derivatives, monotone numerical Hamiltonians and Gauss–Seidel iterations with alternating-direction sweepings. Based on well-developed first order sweeping methods, we design a novel approach to incorporate high order approximations to derivatives into numerical Hamiltonians such that the resulting numerical schemes are formally high order accurate and inherit the fast convergence from the alternating sweeping strategy. Extensive numerical examples verify efficiency, convergence and high order accuracy of the new methods.
Keywordsfast sweeping methods WENO approximation high order accuracy static Hamilton–Jacobi equations Eikonal equations
Unable to display preview. Download preview PDF.
- Dellinger, J., and Symes, W. W. (1997). Anisotropic finite-difference traveltimes using a Hamilton–Jacobi solver, 67th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1786–1789Google Scholar
- Helmsen J., Puckett E., Colella P., Dorr M. (1996) Two new methods for simulating photolithography development in 3d, Proc SPIE, 2726: 253–261Google Scholar
- Qian J., Symes W.W. (2002). An adaptive finite-difference method for traveltime and amplitude. Geophysics 67, 166–176Google Scholar
- Chi-Wang Shu (2004) High Order Numerical Methods for Time Dependent Hamilton–Jacobi Equations, WSPC/Lecture Notes SeriesGoogle Scholar