A Level Set Approach for Computing Discontinuous Solutions of a Class of Hamilton-Jacobi Equations

We introduce two types of finite difference methods to compute the Lsolution [15] and the proper viscosity solution [14] 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 [7]. 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 WENO Local Lax-Friedrichs methods [22]. We verify that our numerical solutions approximate the proper viscosity solutions of [14]. Finally, since the solution of scalar conservation law equations can be Research supported by ONR N00014-97-1-0027, DARPA/NSF VIP grant NSF DMS 9615854 and ARO DAAG 55-98-1-0323 yDepartment of Mathematics, University of California Los Angeles, Los Angeles, California 90095, email:ytsai@math.ucla.edu zDepartment of Mathematics, Hokkaido University, Sapporo 060-0810, Japan, email: giga@math.sci.hokudai.ac.jp xDepartment of Mathematics, University of California Los Angeles, Los Angeles, California 90095, email:sjo@math.ucla.edu

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