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A grid based particle method for solving partial differential equations on evolving surfaces and modeling high order geometrical motion. (English) Zbl 1316.65089

Summary: We develop numerical methods for solving partial differential equations (PDE) defined on an evolving interface represented by the grid based particle method (GBPM) recently proposed in [S. Leung and H. K. Zhao, J. Comput. Phys. 228, No. 8, 2993-3024 (2009; Zbl 1161.65013)]. In particular, we develop implicit time discretization methods for the advection-diffusion equation where the time step is restricted solely by the advection part of the equation. We also generalize the GBPM to solve high order geometrical flows including surface diffusion and Willmore-type flows. The resulting algorithm can be easily implemented since the method is based on meshless particles quasi-uniformly sampled on the interface. Furthermore, without any computational mesh or triangulation defined on the interface, we do not require remeshing or reparametrization in the case of highly distorted motion or when there are topological changes. As an interesting application, we study locally inextensible flows governed by energy minimization. We introduce tension force via a Lagrange multiplier determined by the solution to a Helmholtz equation defined on the evolving interface. Extensive numerical examples are also given to demonstrate the efficiency of the proposed approach.

MSC:

65M75 Probabilistic methods, particle methods, etc. for initial value and initial-boundary value problems involving PDEs
35K55 Nonlinear parabolic equations
65D17 Computer-aided design (modeling of curves and surfaces)
76M25 Other numerical methods (fluid mechanics) (MSC2010)

Citations:

Zbl 1161.65013

Software:

PPM
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Full Text: DOI

References:

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