An embedding technique for the solution of reaction–diffusion equations on algebraic surfaces with isolated singularities

Thomas März, Parousia Rockstroh, Steven J. Ruuth

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

In this paper we construct a parametrization-free embedding technique for numerically evolving reaction-diffusion PDEs defined on algebraic curves that possess an isolated singularity. In our approach, we first desingularize the curve by appealing to techniques from algebraic geometry. We create a family of smooth curves in higher dimensional space that correspond to the original curve by projection. Following this, we pose the analogous reaction-diffusion PDE on each member of this family and show that the solutions (their projection onto the original domain) approximate the solution of the original problem. Finally, we compute these approximants numerically by applying the Closest Point Method which is an embedding technique for solving PDEs on smooth surfaces of arbitrary dimension or codimension, and is thus suitable for our situation. In addition, we discuss the potential to generalize the techniques presented for higher-dimensional surfaces with multiple singularities.
Original languageEnglish (US)
Pages (from-to)911-943
Number of pages33
JournalJournal of Mathematical Analysis and Applications
Volume436
Issue number2
DOIs
StatePublished - Apr 2016
Externally publishedYes

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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