Numerical finite difference approximations of a coupled parabolic system with blow-up

This paper is concerned with the numerical blow-up time for a coupled system of two one-dimensional semilinear parabolic equations with zero Dirichlet boundary conditions. Firstly, we derive the semi-discrete problem and prove that the blow-up solution and numerical blow-up time of the semi-discrete problem are convergent to the theoretical ones, as we refine the space-time grids. Secondly, we derive two fully discrete formulas of standard finite difference methods: the explicit Euler and implicit Euler schemes, with non-fixed time-stepping formula. In addition, we investigate the consistency, stability and convergence of the proposed schemes. Finally, we conduct two numerical experiments to show the accuracy and efficiency of the proposed schemes. Namely, for each experiment, we use the proposed schemes to calculate the numerical blow-up time, error bounds and the numerical order of convergence for blow-up times.