Simple computation of reaction–diffusion processes on point clouds

Colin B. Macdonald, Barry Merriman, Steven J. Ruuth

Research output: Contribution to journalArticlepeer-review

45 Scopus citations


The study of reaction-diffusion processes is much more complicated on general curved surfaces than on standard Cartesian coordinate spaces. Here we show how to formulate and solve systems of reaction-diffusion equations on surfaces in an extremely simple way, using only the standard Cartesian form of differential operators, and a discrete unorganized point set to represent the surface. Our method decouples surface geometry from the underlying differential operators. As a consequence, it becomes possible to formulate and solve rather general reaction-diffusion equations on general surfaces without having to consider the complexities of differential geometry or sophisticated numerical analysis. To illustrate the generality of the method, computations for surface diffusion, pattern formation, excitable media, and bulk-surface coupling are provided for a variety of complex point cloud surfaces.
Original languageEnglish (US)
Pages (from-to)9209-9214
Number of pages6
JournalProceedings of the National Academy of Sciences
Issue number23
StatePublished - May 20 2013
Externally publishedYes

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUK-C1-013-04, KUK-C1-013-04
Acknowledgements: C.B.M. thanks Dr. Chandrasekhar Venkataraman (University of Sussex) for useful discussions on bulk-coupled reaction-diffusion models. The work of C. B. M. was supported by Award KUK-C1-013-04 from King Abdullah University of Science and Technology (KAUST). The work of S.J.R. was partially supported by a Natural Sciences and Engineering Research Council of Canada Discovery Grant and by Award KUK-C1-013-04 from KAUST.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.


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