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 R3. 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.
Scopus Subject Areas
- Applied Mathematics
- Generalized Finite Difference Method
- Localized meshless method
- PDEs on surfaces
- Semi-implicit backward differentiation method