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A local radial basis function differential quadrature semi-discretisation technique for the simulation of time-dependent reaction-diffusion problems

Ram Jiwari (Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, India)
Alf Gerisch (TU Darmstadt, Fachbereich Mathematik, AG Numerik und Wissenschaftliches Rechnen, Darmstadt, Germany)

Engineering Computations

ISSN: 0264-4401

Article publication date: 20 January 2021

Issue publication date: 9 July 2021

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Abstract

Purpose

This paper aims to develop a meshfree algorithm based on local radial basis functions (RBFs) combined with the differential quadrature (DQ) method to provide numerical approximations of the solutions of time-dependent, nonlinear and spatially one-dimensional reaction-diffusion systems and to capture their evolving patterns. The combination of local RBFs and the DQ method is applied to discretize the system in space; implicit multistep methods are subsequently used to discretize in time.

Design/methodology/approach

In a method of lines setting, a meshless method for their discretization in space is proposed. This discretization is based on a DQ approach, and RBFs are used as test functions. A local approach is followed where only selected RBFs feature in the computation of a particular DQ weight.

Findings

The proposed method is applied on four reaction-diffusion models: Huxley’s equation, a linear reaction-diffusion system, the Gray–Scott model and the two-dimensional Brusselator model. The method captured the various patterns of the models similar to available in literature. The method shows second order of convergence in space variables and works reliably and efficiently for the problems.

Originality/value

The originality lies in the following facts: A meshless method is proposed for reaction-diffusion models based on local RBFs; the proposed scheme is able to capture patterns of the models for big time T; the scheme has second order of convergence in both time and space variables and Nuemann boundary conditions are easy to implement in this scheme.

Keywords

Acknowledgements

The work is supported by SERB, India with grant No. YSS/2015/000599 and by DAAD (German Academic Exchange Service).

Citation

Jiwari, R. and Gerisch, A. (2021), "A local radial basis function differential quadrature semi-discretisation technique for the simulation of time-dependent reaction-diffusion problems", Engineering Computations, Vol. 38 No. 6, pp. 2666-2691. https://doi.org/10.1108/EC-05-2020-0291

Publisher

:

Emerald Publishing Limited

Copyright © 2020, Emerald Publishing Limited

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