A grid based particle method for solving partial differential equations on evolving surfaces and modeling high order geometrical motion

  • Authors:

  • Shingyu Leung;John Lowengrub;Hongkai Zhao

  • Affiliations:

  • Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong;Department of Mathematics, University of California at Irvine, Irvine, CA 92697-3875, USA;Department of Mathematics, University of California at Irvine, Irvine, CA 92697-3875, USA

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2011

Quantified Score

Hi-index 31.47

Visualization

Abstract

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, H.K. Zhao, A grid based particle method for moving interface problems, J. Comput. Phys. 228 (2009) 7706-7728]. 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.