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

Thomas März; Parousia Rockstroh; Steven J. Ruuth

doi:10.1016/j.jmaa.2015.11.064

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 journal › Article › peer-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 language	English (US)
Pages (from-to)	911-943
Number of pages	33
Journal	Journal of Mathematical Analysis and Applications
Volume	436
Issue number	2
DOIs	https://doi.org/10.1016/j.jmaa.2015.11.064
State	Published - Apr 2016
Externally published	Yes

Bibliographical note

KAUST Repository Item: Exported on 2021-04-02
Acknowledged KAUST grant number(s): KUK-C1-013-04
Acknowledgements: The authors thank Colin Macdonald for many helpful discussions. The authors also thank the people who contributed to the cp_matrices code (see github.com/cbm755/cp_matrices), in particular Colin Macdonald, Ingrid von Glehn, and Yujia Chen. Thomas März was supported by award KUK-C1-013-04 made by King Abdullah University of Science and Technology (KAUST). Parousia Rockstroh and Steven Ruuth were supported by a grant from NSERC Canada (RGPIN 227823) and by award KUK-C1-013-04 made by King Abdullah University of Science and Technology (KAUST).
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1016/j.jmaa.2015.11.064

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

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.",

author = "Thomas M{\"a}rz and Parousia Rockstroh and Ruuth, {Steven J.}",

note = "KAUST Repository Item: Exported on 2021-04-02 Acknowledged KAUST grant number(s): KUK-C1-013-04 Acknowledgements: The authors thank Colin Macdonald for many helpful discussions. The authors also thank the people who contributed to the cp_matrices code (see github.com/cbm755/cp_matrices), in particular Colin Macdonald, Ingrid von Glehn, and Yujia Chen. Thomas M{\"a}rz was supported by award KUK-C1-013-04 made by King Abdullah University of Science and Technology (KAUST). Parousia Rockstroh and Steven Ruuth were supported by a grant from NSERC Canada (RGPIN 227823) and by award KUK-C1-013-04 made by King Abdullah University of Science and Technology (KAUST). This publication acknowledges KAUST support, but has no KAUST affiliated authors.",

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N2 - 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.

AB - 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.

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

Abstract

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