A least-squares implicit RBF-FD closest point method and applications to PDEs on moving surfaces

A. Petras*, Leevan LING, C. Piret, S. J. Ruuth

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

25 Citations (Scopus)


The closest point method (Ruuth and Merriman (2008) [32]) is an embedding method developed to solve a variety of partial differential equations (PDEs) on smooth surfaces, using a closest point representation of the surface and standard Cartesian grid methods in the embedding space. Recently, a closest point method with explicit time-stepping was proposed that uses finite differences derived from radial basis functions (RBF-FD). Here, we propose a least-squares implicit formulation of the closest point method to impose the constant-along-normal extension of the solution on the surface into the embedding space. Our proposed method is particularly flexible with respect to the choice of the computational grid in the embedding space. In particular, we may compute over a computational tube that contains problematic nodes. This fact enables us to combine the proposed method with the grid based particle method (Leung and Zhao (2009) [37]) to obtain a numerical method for approximating PDEs on moving surfaces. We present a number of examples to illustrate the numerical convergence properties of our proposed method. Experiments for advection–diffusion equations and Cahn–Hilliard equations that are strongly coupled to the velocity of the surface are also presented.

Original languageEnglish
Pages (from-to)146-161
Number of pages16
JournalJournal of Computational Physics
Publication statusPublished - 15 Mar 2019

Scopus Subject Areas

  • Numerical Analysis
  • Modelling and Simulation
  • Physics and Astronomy (miscellaneous)
  • Physics and Astronomy(all)
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics

User-Defined Keywords

  • Closest point method
  • Grid based particle method
  • Least-squares method
  • Partial differential equations on moving surfaces
  • Radial basis functions finite differences (RBF-FD)


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