Efficient Positivity-Preserving NSFD Scheme: Application to Advection-Diffusion-Reaction Equation

Document Type : Research Paper

Authors
Reza Shokri Jahandizi 1 , 2
Mohammad Mehdizadeh Khalsaraei 1
Ali Shokri 1 , 3
Ali Khani 2
1 Department of Mathematics, Faculty of Basic Science, University of Maragheh, Maragheh, Iran
2 Department of Mathematics, Faculty of Basic Science, Azarbaijan Shahid Madani University, Tabriz, Iran
3 Faculty of Mathematics, Statistics and Computer Science, University of Tabriz, Tabriz, Iran
10.22055/jacm.2024.47189.4672
Abstract
Solving problems across a wide range of scientific and engineering domains, encompassing physical, mechanical and biological systems as well as financial markets, necessitates addressing parabolic equations of the advection-diffusion-reaction (ADR) variety. In these multifaceted problems, our primary objectives invariably involve determining the concentration of chemical compounds, the pricing of options, or the scale of populations, all of which inherently possess positive attributes. It is noteworthy, however, that the utilization of conventional methodologies, such as the classical finite difference method, can inadvertently give rise to numerical deficiencies, manifesting as spurious oscillations and negative values in the computed solutions due to truncation errors. By employing the nonstandard finite difference (NSFD) method, an improved finite difference framework is established, one that effectively mitigates the aforementioned issues. Specifically, the proposed NSFD scheme guarantees the positivity of the solutions and effectively eliminates any presence of spurious oscillations within the computed solutions.
Keywords
Advection-diffusion-reaction equations
Fluid dynamics
Nonstandard finite differences
Positivity preserving
Subjects

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Journal of Applied and Computational Mechanics
Volume 11, Issue 3
Summer 2025
Pages 611-624