Fast Sweeping Methods for Factored Anisotropic Eikonal Equations: Multiplicative and Additive Factors

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• 作者：Songting Luo (1) luos@math.msu.edu
  Jianliang Qian (1) qian@math.msu.edu
• 刊名：Journal of Scientific Computing
• 出版年：2012
• 出版时间：August 2012
• 年：2012
• 卷：52
• 期：2
• 页码：360-382
• 全文大小：1.8 MB
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• 作者单位：1. Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
• ISSN：1573-7691

文摘

The viscosity solution of static Hamilton-Jacobi equations with a point-source condition has an upwind singularity at the source, which makes all formally high-order finite-difference scheme exhibit first-order convergence and relatively large errors. To obtain designed high-order accuracy, one needs to treat this source singularity during computation. In this paper, we apply the factorization idea to numerically compute viscosity solutions of anisotropic eikonal equations with a point-source condition. The idea is to factor the unknown traveltime function into two functions, either additively or multiplicatively. One of these two functions is specified to capture the source singularity so that the other function is differentiable in a neighborhood of the source. Then we design monotone fast sweeping schemes to solve the resulting factored anisotropic eikonal equation. Numerical examples show that the resulting monotone schemes indeed yield clean first-order convergence rather than polluted first-order convergence and both factorizations are able to treat the source singularity successfully.