Unconditionally stable methods for Hamilton--Jacobi equations

Authors:
Kenneth Hvistendahl Karlsen;Nils Henrik Risebro
Affiliations:
Department of Mathematics, University of Bergen, Johs, Brunsgt, 12, N-5008 Bergen, Norway;Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, N-0316 Oslo, Norway
Venue:
Journal of Computational Physics
Year:
2002

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Abstract

We present new numerical methods for constructing approximate solutions to the Cauchy problem for Hamilton-Jacobi equations of the form ut + H(Dxu)=0. The methods are based on dimensional splitting and front tracking for solving the associated (non-strictly hyperbolic) system of conservation laws pt + Dx H(p) = 0, where p = Dx u. In particular, our methods depend heavily on a front tracking method for one-dimensional scalar conservation laws with discontinuous coefficients. The proposed methods are unconditionally stable in the sense that the time step is not limited by the space discretization and they can be viewed as "large-time-step" Godunov-type (or front tracking) methods. We present several numerical examples illustrating the main features of the proposed methods. We also compare our methods with several methods from the literature.