The Mathematical Institute, University of Oxford, Eprints Archive

An embedding technique for the solution of reaction-fiffusion equations on algebraic surfaces with isolated singularities

Rockstroh, P. and März, T. and Ruuth, S. J. (2012) An embedding technique for the solution of reaction-fiffusion equations on algebraic surfaces with isolated singularities. Foundations of Computational Mathematics . (Submitted)

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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.

Item Type:Article
Subjects:D - G > General
Research Groups:Oxford Centre for Collaborative Applied Mathematics
ID Code:1674
Deposited By: Peter Hudston
Deposited On:22 Feb 2013 13:32
Last Modified:29 May 2015 19:22

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