An unconditionally positivity preserving scheme for advection–diffusion reaction equations

Parabolic equations with advection, diffusion and reaction terms are used to model many physical and biological systems. In many applications the unknowns represent quantities that cannot be negative such as concentrations of chemical compounds or population sizes. Widely used schemes such as finite differences may produce negative solutions because of truncation errors and may then become unstable. We propose a new scheme that guarantees the positivity of the solutions for arbitrary step sizes. It works for reaction terms that consist of the sum of a positive function and a negative function. We develop it for one advection–diffusion reaction equation in one spatial dimension with constant velocity and diffusion and state how to generalize it. The method is applicable to both advection and diffusion dominated problems. We give some examples from different applications.