Title: The Galerkin reliable scheme for the numerical analysis of the Burgers'-Fisher equation

Authors: Pius W.M. Chin

Addresses: Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Medunsa 0204, Ga-rankuwa, Pretoria, South Africa

Abstract: We consider in this paper, the non-standard finite difference method in the time variable combined with the Galerkin method in the space variables. We use this to study the Burgers'-Fisher equation which is one of the most important nonlinear partial differential equation appearing in various applications such as in fluid dynamics. Existence and uniqueness of the solution of the problem is determined for a given small data in the space L^∞[(0, T); L²(Ω)] ∩ L²[(0, T); H1 0(Ω) ]. The numerical scheme of the problem is designed using the said combination. The proposed scheme is successfully implemented by firstly establishing the stability of the numerical scheme and secondly by determining the estimate for the optimal convergence rate of the numerical solution of the scheme in both the L² as well as H¹-norms. Furthermore, we show that the numerical solution of the scheme preserves the decaying properties of the exact solution of the problem and moreover, the numerical experiments with the help of an example are presented to justify the validity of the results.

Keywords: Burgers'-Fisher equations; fluid dynamics; nonlinear equation; non-standard finite difference method; Galerkin method; optimal rate of convergence.

DOI: 10.1504/PCFD.2021.116530

Progress in Computational Fluid Dynamics, An International Journal, 2021 Vol.21 No.4, pp.234 - 247

Received: 19 Feb 2020
Accepted: 13 Sep 2020
Published online: 27 Jul 2021 *

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