Density-pressure IBVP: Numerical analysis, simulation and cell dynamics in a colonic crypt

• Published in: Applied mathematics and computation 2022-07, Vol.424, p.127037, Article 127037
• Main Authors: Campos, G.C.M., Ferreira, J.A., Romanazzi, G.
• Format: Article
• Language: English
• Subjects: Cell dynamics; Colonic crypt; Coupled system of elliptic and parabolic equations; Finite difference methods; Mixed derivatives
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2022.127037

•Numerical method for convection-diffusion-reaction equations with mixed derivatives.•Semi-discrete finite difference method considered in nonuniform rectangular grid.•Proof of Stability and Second convergence order with respect to a discrete H1-norm.•Application of the method for simulating the cell dynamics in a colonic crypt. This paper proposes a finite difference method for a differential-algebraic system that couples an elliptic equation with a convection-diffusion-reaction equation with mixed derivatives. Herein the parabolic equation depends on the gradient of the solution of the elliptic equation. If the numerical approximation for this gradient presents lower accuracy then the numerical approximations for the solution of the parabolic equation can be deteriorated. In order to get a second order approximation for the solution of the parabolic equation, the challenges that we have to face are the construction of the right discretizations of the elliptic and parabolic equations that lead to a second order approximation for the gradient as well as the development of the numerical analysis of the proposed method. The differential system analysed can be used to describe the cell density and pressure dynamics in colonic crypts. Numerical simulations of cell dynamics in a crypt of the human colon during the formation of a micro-adenoma are presented.