Article

Journal of Scientific Computing

, Volume 35, Issue 2, pp 219-240

First online:

Level Set Equations on Surfaces via the Closest Point Method

  • Colin B. MacdonaldAffiliated withDepartment of Mathematics, Simon Fraser University
  • , Steven J. RuuthAffiliated withDepartment of Mathematics, Simon Fraser University Email author 

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Abstract

Level set methods have been used in a great number of applications in ℝ2 and ℝ3 and it is natural to consider extending some of these methods to problems defined on surfaces embedded in ℝ3 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.

Keywords

Closest Point Method Level set methods Partial differential equations Implicit surfaces WENO schemes WENO interpolation