Developing Higher-Order Unconditionally Positive Finite Difference Methods for the Advection Diffusion Reaction Equations

Author:

Ndou Ndivhuwo12ORCID,Dlamini Phumlani1ORCID,Jacobs Byron Alexander1ORCID

Affiliation:

1. Department of Mathematics and Applied Mathematics, University of Johannesburg, P.O. Box 524, Auckland Park, Johannesburg 2006, South Africa

2. Department of Mathematical and Computational Sciences, University of Venda, Private Bag X5050, Thohoyandou 0950, South Africa

Abstract

This study introduces the higher-order unconditionally positive finite difference (HUPFD) methods to solve the linear, nonlinear, and system of advection–diffusion–reaction (ADR) equations. The stability and consistency of the developed methods are analyzed, which are necessary and sufficient for the numerical approach to converge to the exact solution. The problem under consideration is of the Cauchy type, and hence, Von Neumann stability analysis is used to analyze the stability of the proposed schemes. The HUPFD’s efficacy and efficiency are investigated by calculating the error, convergence rate, and computing time. For validation purposes, the higher-order unconditionally positive finite difference solutions are compared to analytical calculations. The numerical results demonstrate that the proposed methods produce accurate solutions to solve the advection diffusion reaction equations. The results also show that increasing the order of the unconditionally positive finite difference leads an implicit scheme that is conditionally stable and has a higher order of accuracy with respect to time and space.

Funder

University of Johannesburg

University of Venda

Publisher

MDPI AG

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