Numerical Finite-Difference Approximations of a Coupled Reaction-Diffusion System with Gradient Terms

This study focuses on the derivation of explicit and implicit finite difference formulas.The objective of this study is to derive an estimation of the blow-up time for a coupled reactiontions. Furthermore, an examination is conducted on the consistency, stability, and convergence of the proposed schemes. Additionally, the study presents two numerical experiments. In each instance, the numerical blow-up time is calculated benefit the suggested methodologies, employing varying space steps and non-fixed time-stepping. The numerical findings obtained demonstrate that the blow-up time sequence exhibits convergence as the space step decreases. Moreover, the numerical orders of convergence for the blow-up time goes well with the theoretical orders observed in the numerical solutions.