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.

On a Framework for the Stability and Convergence Analysis of Discrete Schemes for Nonstationary Nonlocal Problems of Parabolic Type

“…We note that very interesting results were obtained in [25], where the authors constructed a non-standard unconditionally positive finite-difference scheme and applied it to solve three test problems for the problem (28). We note that this scheme has first-order accuracy in time and second-order accuracy in space.…”

“…where U n− 1 2 = 0.5 U n + U n−1 . In order to efficiently compute the solution U n , we write the CN scheme (25) in the following form:…”

“…The stability factor of the CN scheme (25) has a more complicated dependence on the large eigenvalues of A h , and this factor changes the sign and converges to (−1). Thus, it is most important to investigate the stability of the scheme (27).…”

“…The mathematical model is generalized to include a nonlocal diffusion operator. The governing two-dimensional mathematical model is based on the Allen-Cahn equation ( [25,26]):…”

“…In order to illustrate the performance of the proposed scheme, we present some results on the simulation of biofilm on medical implants. This problem was also investigated in [25,27]. We considered an initial profile with the density function given as u(x, y, 0) = 2 ∑ p=1 C p e −w p X−X p 2 , where X = (x, y), C 1 = 0.68, C 2 = 0.65, w 1 = 60, w 2 = 100, X 1 = (0.35, 0.60), X 2 = (0.48, 0.45).…”