High-order semi-discrete central-upwind schemes for multi-dimensional Hamilton--Jacobi equations

Authors:
Steve Bryson;Doron Levy
Affiliations:
Program in Scientific Computing and Computational Mathematics, Stanford University and the NASA Advanced Supercomputing Division, NASA Ames Research Center, Moffett Field, CA;Department of Mathematics, Stanford University, Stanford, CA
Venue:
Journal of Computational Physics
Year:
2003

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Abstract

We present the first fifth-order, semi-discrete central-upwind method for approximating solutions of multi-dimensional Hamilton-Jacobi equations. Unlike most of the commonly used high-order upwind schemes, our scheme is formulated as a Godunov-type scheme. The scheme is based on the fluxes of Kurganov-Tadmor and Kurganov-Noelle-Petrova, and is derived for an arbitrary number of space dimensions. A theorem establishing the monotonicity of these fluxes is provided. The spatial discretization is based on a weighted essentially non-oscillatory reconstruction of the derivative. The accuracy and stability properties of our scheme are demonstrated in a variety of examples. A comparison between our method and other fifth-order schemes for Hamilton-Jacobi equations shows that our method exhibits smaller errors without any increase in the complexity of the computations.