Enhanced Unconditionally Positive Finite Difference Method for Advection–Diffusion–Reaction Equations

Ndivhuwo Ndou, Phumlani Dlamini and Byron Alexander Jacobs
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Ndivhuwo Ndou: Department of Mathematics and Applied Mathematics, University of Johannesburg, Auckland Park, Johannesburg 2006, South Africa
Phumlani Dlamini: Department of Mathematics and Applied Mathematics, University of Johannesburg, Auckland Park, Johannesburg 2006, South Africa
Byron Alexander Jacobs: Department of Mathematics and Applied Mathematics, University of Johannesburg, Auckland Park, Johannesburg 2006, South Africa

Mathematics, 2022, vol. 10, issue 15, 1-18

Abstract: In this study, we develop the enhanced unconditionally positive finite difference method (EUPFD), and use it to solve linear and nonlinear advection–diffusion–reaction (ADR) equations. This method incorporates the proper orthogonal decomposition technique to the unconditionally positive finite difference method (UPFD) to reduce the degree of freedom of the ADR equations. We investigate the efficiency and effectiveness of the proposed method by checking the error, convergence rate, and computational time that the method takes to converge to the exact solution. Solutions obtained by the EUPFD were compared with the exact solutions for validation purposes. The agreement between the solutions means the proposed method effectively solved the ADR equations. The numerical results show that the proposed method greatly improves computational efficiency without a significant loss in accuracy for solving linear and nonlinear ADR equations.

Keywords: proper orthogonal decomposition; unconditionally positive finite difference method; advection–diffusion–reaction equations; enhanced unconditionally positive finite difference method (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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Citations: View citations in EconPapers (3)

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