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Volume 28, Issue 4
A Compact Upwind Second Order Scheme for the Eikonal Equation

J.-D.Benamou, Songting Luo & Hongkai Zhao

DOI: 10.4208/jcm.1003-m0014

J. Comp. Math., 28 (2010), pp. 489-516.

Published online: 2010-08

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  • 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.

  • Keywords

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

  • AMS Subject Headings

35L60, 65N06, 65N12, 65N15

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-28-489, author = {J.-D.Benamou, Songting Luo and Hongkai Zhao}, title = {A Compact Upwind Second Order Scheme for the Eikonal Equation}, journal = {Journal of Computational Mathematics}, year = {2010}, volume = {28}, number = {4}, pages = {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.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1003-m0014}, url = {http://global-sci.org/intro/article_detail/jcm/8534.html} }
TY - JOUR T1 - A Compact Upwind Second Order Scheme for the Eikonal Equation AU - J.-D.Benamou, Songting Luo & Hongkai Zhao JO - Journal of Computational Mathematics VL - 4 SP - 489 EP - 516 PY - 2010 DA - 2010/08 SN - 28 DO - http://doi.org/10.4208/jcm.1003-m0014 UR - https://global-sci.org/intro/article_detail/jcm/8534.html KW - Eikonal equation, Upwind scheme, Hamilton-Jacobi, Viscosity Solution, Sweeping method. AB -

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.

J.-D.Benamou, Songting Luo and Hongkai Zhao. (2010). A Compact Upwind Second Order Scheme for the Eikonal Equation. Journal of Computational Mathematics. 28 (4). 489-516. doi:10.4208/jcm.1003-m0014
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