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

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.