A level set approach for computing discontinuous solutions of Hamilton-Jacobi equations
A level set approach for computing discontinuous solutions of Hamilton-Jacobi equations
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DOI：
10.1090/s0025-5718-02-01438-2
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发表时间：
2003
期刊：
影响因子：
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通讯作者：
Y. Tsai;Y. Giga;S. Osher
中科院分区：
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，分别计算了一类Hamilton-Jacobi方程半间断解的L-解和第二作者最近提出的适当粘性解。将解的图形看作是一维高阶连续函数的零水平曲线，我们可以用Crandall和Lions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。这些数值方法推广到高阶精度加权ENO局部Lax-Friedrichs方法最近开发的江和彭。我们验证了我们的数值解近似适当的粘度解决方案，由第二作者在最近的北海道大学预印本。最后，由于标量守恒律方程的解可以使用现有的数值技术构造，我们用它来验证我们的数值解近似熵解。
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.