A closest point penalty method for evolution equations on surfaces - von Glehn, I (2014)
This thesis introduces and analyses a numerical method for solving time-dependent partial differential equations (PDEs) on surfaces. This method is based on the closest point method, and solves the surface PDE by solving a suitably chosen equation in a band surrounding the surface. As it uses an implicit closest point representation of the surface, the method has the advantages of being simple to implement for very general surfaces, and amenable to discretization with a broad class of numerical schemes. The method proposed in this work introduces a new equation in the embedding space, which satisfies a key consistency property with the surface PDE. Rather than alternating between explicit time-steps and re-extensions of the surface function as in the original closest point method, we investigate an alternative approach, in which a single equation can be solved throughout the embedding space, without separate extension steps. This is achieved by creating a modified embedding equation with a penalty term, which enforces a constraint on the solution. The resulting equation admits a method of lines discretization, and can therefore be discretized with implicit or explicit time-stepping schemes, and analysed with standard techniques. The method can be formulated in a straightforward way for a large class of problems, including equations featuring variable coefficients, higher-order terms or nonlinearities. The effectiveness of the method is demonstrated with a range of examples, drawing from applications involving curvature-dependent diffusion and systems of reaction-diffusion equations, as well as equations arising in PDE-based image processing on surfaces.