A level set approach for computing discontinuous solutions of Hamilton-Jacobi equations. (English) Zbl 1013.65088
Summary: 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 (HJ) 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 M. G. Crandall and P.-L. Lions [Trans. Am. Math. Soc. 277, 1-42 (1983; Zbl 0599.35024)]. 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 essentially non-oscillatory local Lax-Friedrichs methods as developed recently by G.-S. Jiang and D. Peng [SIAM J. Sci. Comput. 21, No. 6, 2126-2143 (2000; Zbl 0957.35014)]. We verify that our numerical solutions approximate the proper viscosity solutions obtained by Y. Giga [Commun. Pure Appl. Math. 55, No. 4, 431-480 (2002; Zbl 1028.35044)].
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.
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 essentially non-oscillatory local Lax-Friedrichs methods as developed recently by G.-S. Jiang and D. Peng [SIAM J. Sci. Comput. 21, No. 6, 2126-2143 (2000; Zbl 0957.35014)]. We verify that our numerical solutions approximate the proper viscosity solutions obtained by Y. Giga [Commun. Pure Appl. Math. 55, No. 4, 431-480 (2002; Zbl 1028.35044)].
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.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 70H20 | Hamilton-Jacobi equations in mechanics |
| 35L60 | First-order nonlinear hyperbolic equations |
| 49L25 | Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games |
Keywords:
Hamilton-Jacobi equations; singular diffusion; level sets; discontinuous solutions; finite difference methods; \(L\)-solution; viscosity solution; conservation laws; Lax-Friedrichs methodsReferences:
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