A localized extrinsic collocation method for Turing pattern formations on surfaces

In this paper, we give our first attempt to implement a localized collocation method, namely Generalized Finite Difference Method (GFDM), for the Turing patterns formation problems on smooth, closed, connected surfaces of codimension one embedded in R-3. Based on projections from surface differential operators to Euclidean differential operators, the surface PDEs in extrinsic form are given explicitly and could be solved directly by GFDM only using a set of collocation points distributed on surfaces. A sparse system formed from localization scheme makes it efficient for solving long time evolution Turing patterns formation problems. Numerical demonstrations including convergence test, Turing spot and stripe problems are provided to illustrate its potentiality. (C) 2021 Elsevier Ltd. All rights reserved.

Automated summary

This paper presents our first attempt to implement a localized collocation method, GFDM, for the Turing patterns formation problems on smooth, closed, connected surfaces of codimension one in R-3. By projecting surface differential operators to Euclidean differential operators, explicit surface PDEs are given and solved using a set of collocation points distributed on surfaces. The sparse system formed from the localization scheme efficiently solves long time evolution Turing patterns formation problems, as demonstrated through numerical tests on convergence, Turing spot and stripe problems.