Solving 3-D Gray–Scott Systems with Variable Diffusion Coefficients on Surfaces by Closest Point Method with RBF-FD

Marzieh Raei, Salvatore Cuomo, Giovanni Colecchia and Gerardo Severino
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Marzieh Raei: Department of Applied Mathematics, Malek Ashtar University of Technology, Tehran 158751774, Iran
Salvatore Cuomo: Scuola Politecnica e delle Scienze di Base, University of Naples Federico II, 80138 Napoli, Italy
Giovanni Colecchia: Scuola Politecnica e delle Scienze di Base, University of Naples Federico II, 80138 Napoli, Italy
Gerardo Severino: Scuola di Agraria e Medicina Veterinaria, University of Naples Federico II, 80138 Napoli, Italy

Mathematics, 2021, vol. 9, issue 9, 1-20

Abstract: The Gray–Scott (GS) model is a non-linear system of equations generally adopted to describe reaction–diffusion dynamics. In this paper, we discuss a numerical scheme for solving the GS system. The diffusion coefficients of the model are on surfaces and they depend on space and time. In this regard, we first adopt an implicit difference stepping method to semi-discretize the model in the time direction. Then, we implement a hybrid advanced meshless method for model discretization. In this way, we solve the GS problem with a radial basis function–finite difference (RBF-FD) algorithm combined with the closest point method (CPM). Moreover, we design a predictor–corrector algorithm to deal with the non-linear terms of this dynamic. In a practical example, we show the spot and stripe patterns with a given initial condition. Finally, we experimentally prove that the presented method provides benefits in terms of accuracy and performance for the GS system’s numerical solution.

Keywords: radial basis function; reaction–diffusion; kernel methods; finite difference (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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