Level set equations on surfaces via the closest point method. (English) Zbl 1203.65143
Summary: Level set methods have been used in a great number of applications in \(\mathbb R^{2}\) and \(\mathbb R^{3}\) and it is natural to consider extending some of these methods to problems defined on surfaces embedded in \(\mathbb R^{3}\) 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 [St. J. Ruuth and B. Merriman [J. Comput. Phys. 227, No. 3, 1943–1961 (2008; Zbl 1134.65058)]. 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.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
closest point method; level set methods; partial differential equations; implicit surfaces; WENO schemes; WENO interpolationCitations:
Zbl 1134.65058References:
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