A Fast Sweeping Method for Eikonal Equations on Implicit Surfaces

Abstract

We propose a computational efficient yet simple numerical algorithm to solve the surface eikonal equation on general implicit surfaces. The method is developed based on the embedding idea and the fast sweeping methods. We first approximate the solution to the surface eikonal equation by the Euclidean weighted distance function defined in a tubular neighbourhood of the implicit surface, and then apply the efficient fast sweeping method to numerically compute the corresponding viscosity solution. Unlike some other embedding methods which require the radius of the computational tube satisfies $h=O(\varDelta x^{\gamma })$ for some $\gamma <1$, our approach allows $h=O(\varDelta x)$. This implies that the total number of grid points in the computational tube is optimal and is given by $O(\varDelta x^{1-d})$ for a co-dimensional one surface in ${\mathbb {R}}^d$. The method can be easily extended to general static Hamilton–Jacobi equation defined on implicit surfaces. Numerical examples will demonstrate the robustness and convergence of the proposed approach.