Numerical blow-up analysis of linearly implicit Euler method for nonlinear parabolic integro-differential equations

The linearly implicit Euler method for nonlinear parabolic integro-differential equations (NPIDEs) on bounded domains is consideblue by approximating the local effects implicitly and nonlocal effects explicitly. Based on Nakagawa's criteria, a suitable adaptive time-stepping strategy is introduced by the discrete energy instead of the infinite norm. With the help of lower discrete energies, the finite blow-up behaviors are replicated for any positive solution. Numerical simulations are carried out to examine the effectiveness of our blowup analysis, which also motivate furthermore to prove that a global numerical solution exists for certain NPIDEs with a weakly singular kernel.