Finite Difference Scheme for One-dimensional Coupled Parabolic System with Blow-up

This study aims to find an efficient technique to estimate the blow-up time (BUT) of one-dimensional semilinear coupled parabolic systems. Firstly, a fully discrete finite difference formula is derived with a nonfixed time-stepping formula, based on the Crank-Nicolson method. In addition, the consistency, stability, and convergence of the proposed scheme are considered. Secondly, two numerical experiments are presented. For each experiment, we apply the proposed scheme to calculate the numerical blow-up time, error bounds, and the numerical order of convergence for blow-up times. The results obtained show that the proposed C.N scheme is consistent with the system considered. However, it is conditionally stable. In addition, the Crank-Nicolson scheme converges in the stability region and achieves first- and second-order accuracy in temporal and spatial dimensions, respectively. Moreover, the suggested nonfixed-stepping formula plays an important role in avoiding any possible instability that may appear near the blow-up point. In addition, it helps to increase the order of numerical convergence. Furthermore, the numerical experiments demonstrate that the numerical blow-up simultaneously occurs at only the center point. Finally, the numerical blow-up time sequence is convergent, as the space step is small enough. In addition, the numerical order of convergence for the blow-up time agrees well with the theoretical order of convergence of the proposed scheme.