Framework for solving dynamics of Ca²⁺ ion concentrations in liver cells numerically: Analysis of a non‐negativity‐preserving non‐standard finite‐difference method

Abstract

In this article, we consider the numerical approximation of dynamics of calcium ion concentrations in liver cells by a non‐linear first‐order system of ordinary differential equations. At first, we summarize some analytical results of the time‐continuous problem formulation which are of importance to numerically solving this system. We continue with the presentation of our numerical framework which contains two ingredients—an optimization approach for determining the unique equilibrium state and, as our main contribution, a non‐standard finite‐difference scheme for numerical approximation of the time‐discretized dynamical system. We prove that our proposed scheme conserves non‐negativity, and we show convergence towards the time‐continuous unique solution. Conclusively, we stress our theoretical findings by numerical examples.