The alternating evolution methods for first order nonlinear partial differential equations

In this paper, we present a brief survey of the recent developments in alternating evolution (AE) methods for numerical computation of first order partial differential equations, with hyperbolic conservation laws and Hamilton-Jacobi equations as two canonical examples. The main difficulty of such computation arises from the nonlinearity of the model, making it necessary to incorporate an appropriate amount of numerical viscosity to capture the entropy/viscosity solution as physically relevant solutions. The alternating evolution method is based on the AE system of the original PDEs, the discretization technique ranges from finite difference, finite volume and the discontinuous Galerkin methods. In all these cases, the AE solver can produce accurate solutions with equal computational time than the traditional solvers. In particular, the AE formulation allows the same discontinuous Galerkin discretization for both conservative and non-conservative PDEs under consideration. In order to make the presentation more concise and to highlight the main ideas of the algorithm, we use simplified models to describe the details of the AE method. Sample simulation results on a few models are also given.

Keywords

conservation laws, Hamilton-Jacobi equations, alternating evolution methods, Lax-Friedrichs

2010 Mathematics Subject Classification

35K15, 65M15, 65M60, 76R50

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Published 3 June 2014