Level Set Equations on Surfaces via the Closest Point Method

  • Authors:

  • Colin B. Macdonald;Steven J. Ruuth

  • Affiliations:

  • Department of Mathematics, Simon Fraser University, Burnaby, Canada V5A-1S6;Department of Mathematics, Simon Fraser University, Burnaby, Canada V5A-1S6

  • Venue:
  • Journal of Scientific Computing
  • Year:
  • 2008

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Abstract

Level set methods have been used in a great number of applications in 驴2 and 驴3 and it is natural to consider extending some of these methods to problems defined on surfaces embedded in 驴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.