Legendre-transform-based fast sweeping methods for static Hamilton-Jacobi equations on triangulated meshes

Authors:
Chiu-Yen Kao;Stanley Osher;Jianliang Qian
Affiliations:
Department of Mathematics, The Ohio State University, Columbus, OH 43210, United States;Department of Mathematics, University of California Los Angeles, Los Angeles, CA 90095, United States;Department of Mathematics, Michigan State University, East Lansing, MI 48824, United States
Venue:
Journal of Computational Physics
Year:
2008

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Abstract

We propose a new sweeping algorithm which utilizes the Legendre transform of the Hamiltonian on triangulated meshes. The algorithm is a general extension of the previous proposed algorithm by Kao et al. [C.Y. Kao, S.J. Osher, Y.-H. Tsai, Fast sweeping method for static Hamilton-Jacobi equations, SIAM J. Numer. Anal. 42 (2005) 2612-2632]. The algorithm yields the numerical solution at a grid point using only its one-ring neighboring grid values and is easy to implement numerically. The minimization that is related to the Legendre transform in the sweeping algorithm can either be solved analytically or numerically. The scheme is shown to be monotone and consistent. We illustrate the efficiency and accuracy of the new method with several numerical examples in two and three dimensions.