English

An embedded method-of-lines approach to solving partial differential equations on surfaces

Numerical Analysis 2013-07-23 v1

Abstract

We introduce a method-of-lines formulation of the closest point method, a numerical technique for solving partial differential equations (PDEs) defined on surfaces. This is an embedding method, which uses an implicit representation of the surface in a band containing the surface. We define a modified equation in the band, obtained in a straightforward way from the original evolution PDE, and show that the solutions of this equation are consistent with those of the surface equation. The resulting system can then be solved with standard implicit or explicit time-stepping schemes, and the solutions in the band can be restricted to the surface. Our derivation generalizes existing formulations of the closest point method and is amenable to standard convergence analysis.

Keywords

Cite

@article{arxiv.1307.5657,
  title  = {An embedded method-of-lines approach to solving partial differential equations on surfaces},
  author = {Ingrid von Glehn and Thomas März and Colin B. Macdonald},
  journal= {arXiv preprint arXiv:1307.5657},
  year   = {2013}
}

Comments

19 pages, 17 figures

R2 v1 2026-06-22T00:55:19.062Z