A Level Set Approach for Computing Discontinuous Solutions of a Class of Hamilton-Jacobi Equations
A Level Set Approach for Computing Discontinuous Solutions of a Class of Hamilton-Jacobi Equations
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发表时间：
2001-08
影响因子：
2
通讯作者：
Y. Tsai;Y. Giga;S. Osher
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文献类型：
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作者：
Y. Tsai;Y. Giga;S. Osher
本文介绍了两类有限差分方法，分别计算了第二作者最近提出的一类Hamilton-Jacobi方程半间断解的L解[15]和适当粘性解[14]。把解的图形看作是一个高一维连续函数的零水平曲线，我们可以用Crandall和Lions [7]引入的粘性理论来处理相应的水平集方程。然而，我们需要在分析和数值上特别注意，以防止零水平曲线翻转，从而可以将其解释为函数的图形。我们展示了我们的Lax-Friedrichs型数值方法计算的L-解决方案，使用其原来的水平集制定。此外，我们耦合我们的数值方法与奇异扩散项，这是必不可少的计算解决方案，更一般的一类HJ方程，包括守恒律。有了这种奇异粘性，我们的数值方法不需要方程的发散结构，并适用于更一般的方程发展的冲击比守恒律。使用韦诺局部Lax-Friedrichs方法将这些数值方法推广到更高阶精度[22]。我们验证了我们的数值解近似于[14]的适当粘性解。最后，由于标量守恒律方程的解可以由ONR N 00014 -97-1-0027，DARPA/NSF VIP grant NSF DMS 9615854和ARO DAAG 55-98-1-0323 y数学系，加州大学洛杉矶分校，加州90095，email:ytsai@math.ucla.edu z数学系，北海道大学，札幌060-0810，日本，电子邮件：giga@math.sci.hokudai.ac.jp x数学系，加州大学洛杉矶分校，加州90095，email:sjo@math.ucla.edu
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