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

Shing-Yu Leung John Lowengrub Hongkai Zhao

Numerical Analysis and Scientific Computing mathscidoc:1912.43180

Journal of Computational Physics, 230, (7), 2540-2561, 2011.4
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) 77067728]. In particular, we develop implicit time discretization methods for the advectiondiffusion 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
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@inproceedings{shing-yu2011a,
  title={A grid based particle method for solving partial differential equations on evolving surfaces and modeling high order geometrical motion},
  author={Shing-Yu Leung, John Lowengrub, and Hongkai Zhao},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191221112800508123740},
  booktitle={Journal of Computational Physics},
  volume={230},
  number={7},
  pages={2540-2561},
  year={2011},
}
Shing-Yu Leung, John Lowengrub, and Hongkai Zhao. A grid based particle method for solving partial differential equations on evolving surfaces and modeling high order geometrical motion. 2011. Vol. 230. In Journal of Computational Physics. pp.2540-2561. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191221112800508123740.
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