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A Compact Upwind Second Order Scheme for the Eikonal Equation

A Compact Upwind Second Order Scheme for the Eikonal Equation
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Year:    2010

Journal of Computational Mathematics, Vol. 28 (2010), Iss. 4 : pp. 489–516

Abstract

We present a compact upwind second order scheme for computing the viscosity solution of the Eikonal equation. This new scheme is based on:
1. the numerical observation that classical first order monotone upwind schemes for the Eikonal equation yield numerical upwind gradient which is also first order accurate up to singularities;
2. a remark that partial information on the second derivatives of the solution is known and given in the structure of the Eikonal equation and can be used to reduce the size of the stencil.  
We implement the second order scheme as a correction to the well known sweeping method but it should be applicable to any first order monotone upwind scheme. Care is needed to choose the appropriate stencils to avoid instabilities. Numerical examples are presented.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/jcm.1003-m0014

Journal of Computational Mathematics, Vol. 28 (2010), Iss. 4 : pp. 489–516

Published online:    2010-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    28

Keywords:    Eikonal equation Upwind scheme Hamilton-Jacobi Viscosity Solution Sweeping method.

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