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Unconditionally stable methods for Hamilton-Jacobi equations. (English) Zbl 1143.65365
Summary: We present new numerical methods for constructing approximate solutions to the Cauchy problem for Hamilton-Jacobi equations of the form $$ut+H(D_xu)=0$$. The methods are based on dimensional splitting and front tracking for solving the associated (non-strictly hyperbolic) system of conservation laws $$pt+D_xH(p)=0$$, where $$p=D_xu$$. 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.

##### MSC:
 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 35F25 Initial value problems for nonlinear first-order PDEs
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##### References:
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