Unconditionally positive NSFD and classical finite difference schemes for biofilm formation on medical implant using Allen-Cahn equation

Abstract

The study of biofilm formation is becoming increasingly important. Microbes that produce biofilms have complicated impact on medical implants. In this paper, we construct an unconditionally positive non-standard finite difference scheme for a mathematical model of biofilm formation on a medical implant. The unknowns in many applications reflect values that cannot be negative, such as chemical component concentrations or population numbers. The model employed here uses the bistable Allen-Cahn partial differential equation, which is a generalization of Fisher’s equation. We study consistency and convergence of the scheme constructed. We compare the performance of our scheme with a classical finite difference scheme using four numerical experiments. The technique used in the construction of unconditionally positive method in this study can be applied to other areas of mathematical biology and sciences. The results here elaborate the benefits of the non-standard approximations over the classical approximations in practical applications.

Keywords

• biofilm
• finite difference method
• advection-diffusion-reaction
• positivity
• boundedness
• 65m06
• 5h05
• 68w10
• 92-10