Level Set Equations on Surfaces via the Closest Point Method

Level set methods have been used in a great number of applications in ℝ 2 and ℝ 3 and it is natural to consider extending some of these methods to problems defined on surfaces embedded in ℝ 3 or higher dimensions. In this paper we consider the treatment of level set equations on surfaces via a recent technique for solving partial differential equations (PDEs) on surfaces, the Closest Point Method (Ruuth and Merriman, J. Comput. Phys. 227(3):1943–1961, [ 2008 ]). Our main modification is to introduce a Weighted Essentially Non-Oscillatory (WENO) interpolation step into the Closest Point Method. This, in combination with standard WENO for Hamilton–Jacobi equations, gives high-order results (up to fifth-order) on a variety of smooth test problems including passive transport, normal flow and redistancing. The algorithms we propose are straightforward modifications of standard codes, are carried out in the embedding space in a well-defined band around the surface and retain the robustness of the level set method with respect to the self-intersection of interfaces. Numerous examples are provided to illustrate the flexibility of the method with respect to geometry.