Positivity-preserving nonstandard finite difference Schemes for simulation of advection-diffusion reaction equations

Systems in which reaction terms are coupled to diffusion and advection transports arise in a wide range of chemical engineering applications, physics, biology and environmental. In these cases, the components of the unknown can denote concentrations or population sizes which represent quantities and they need to remain positive. Classical finite difference schemes may produce numerical drawbacks such as spurious oscillations and negative solutions because of truncation errors and may then become unstable. we propose a new scheme that guarantees a smooth numerical solution, free of spurious oscillations and satisfies the positivity requirement, as is demanded for the advection-diffusion reaction equations. The method is applicable to both advection and diffusion dominated problems. We give some examples from different applications.