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

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Title: A level set approach for computing discontinuous solutions of a class of Hamilton-Jacobi equations
Authors: Tsai, Y.-H. R Browse this author
Giga, Y. Browse this author
Osher, S. Browse this author
Issue Date: Aug-2001
Publisher: Department of Mathematics, Hokkaido University
Journal Title: Hokkaido University Preprint Series in Mathematics
Volume: 534
Start Page: 1
End Page: 30
Abstract: We introduce two types of finite difference methods to compute the L­solution [14] and the proper viscosity solution [13] 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 [21]. We verify that our numerical solutions approximate the proper viscosity solutions of [ 13]. 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.
Type: bulletin (article)
Appears in Collections:理学院・理学研究院 (Graduate School of Science / Faculty of Science) > Hokkaido University Preprint Series in Mathematics

Submitter: 数学紀要登録作業用

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