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

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.