TY - JOUR
T1 - An embedding technique for the solution of reaction–diffusion equations on algebraic surfaces with isolated singularities
AU - März, Thomas
AU - Rockstroh, Parousia
AU - Ruuth, Steven J.
N1 - 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.
PY - 2016/4
Y1 - 2016/4
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.
UR - http://hdl.handle.net/10754/668477
UR - https://linkinghub.elsevier.com/retrieve/pii/S0022247X15011373
UR - http://www.scopus.com/inward/record.url?scp=84955174027&partnerID=8YFLogxK
U2 - 10.1016/j.jmaa.2015.11.064
DO - 10.1016/j.jmaa.2015.11.064
M3 - Article
SN - 0022-247X
VL - 436
SP - 911
EP - 943
JO - Journal of Mathematical Analysis and Applications
JF - Journal of Mathematical Analysis and Applications
IS - 2
ER -