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A level set approach for computing discontinuous solutions of Hamilton-Jacobi equations

Authors: Yen-Hsi Richard Tsai, Yoshikazu Giga and Stanley Osher
Journal: Math. Comp. 72 (2003), 159-181
MSC (2000): Primary 65Mxx, 35Lxx; Secondary 70H20
Published electronically: August 13, 2002
MathSciNet review: 1933817
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Abstract: We introduce two types of finite difference methods to compute the L-solution and the proper viscosity solution recently proposed by the second author for semi-discontinuous solutions to a class of Hamilton-Jacobi equations. By regarding the graph of the solution as the zero level curve of a continuous function in one dimension higher, we can treat the corresponding level set equation using the viscosity theory introduced by Crandall and Lions. However, we need to pay special attention both analytically and numerically to prevent the zero level curve from overturning so that it can be interpreted as the graph of a function. We demonstrate our Lax-Friedrichs type numerical methods for computing the L-solution using its original level set formulation. In addition, we couple our numerical methods with a singular diffusive term which is essential to computing solutions to a more general class of HJ equations that includes conservation laws. With this singular viscosity, our numerical methods do not require the divergence structure of equations and do apply to more general equations developing shocks other than conservation laws. These numerical methods are generalized to higher order accuracy using weighted ENO local Lax-Friedrichs methods as developed recently by Jiang and Peng. We verify that our numerical solutions approximate the proper viscosity solutions obtained by the second author in a recent Hokkaido University preprint. Finally, since the solution of scalar conservation law equations can be constructed using existing numerical techniques, we use it to verify that our numerical solution approximates the entropy solution.

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Additional Information

Yen-Hsi Richard Tsai
Affiliation: Department of Mathematics, University of California Los Angeles, Los Angeles, California 90095

Yoshikazu Giga
Affiliation: Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan

Stanley Osher
Affiliation: Department of Mathematics, University of California Los Angeles, Los Angeles, California 90095

Keywords: Hamilton-Jacobi equations, singular diffusion, level sets
Received by editor(s): March 7, 2001
Published electronically: August 13, 2002
Additional Notes: The first and the third authors are supported by ONR N00014-97-1-0027, DARPA/NSF VIP grant NSF DMS 9615854 and ARO DAAG 55-98-1-0323.
Article copyright: © Copyright 2002 American Mathematical Society