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

Author

Listed:
  • 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)

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.

Suggested Citation

  • Marzieh Raei & Salvatore Cuomo & Giovanni Colecchia & Gerardo Severino, 2021. "Solving 3-D Gray–Scott Systems with Variable Diffusion Coefficients on Surfaces by Closest Point Method with RBF-FD," Mathematics, MDPI, vol. 9(9), pages 1-20, April.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:9:p:924-:d:540563
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    References listed on IDEAS

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    1. Gerardo Severino & Alessandro Santini & Valeria Marina Monetti, 2009. "Modelling Water Flow water flow and Solute Transport in Heterogeneous Unsaturated Porous Media," Springer Optimization and Its Applications, in: Panos M. Pardalos & Petraq J. Papajorgji (ed.), Advances in Modeling Agricultural Systems, pages 361-383, Springer.
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