Blow-up numerical solutions for a convective reaction-diffusion equation

We consider finite difference solutions of the 1-dim convective reaction-diffusion equation u(t) = u(xx) + alpha(u(m))(x) + u(beta) (alpha > 0, beta >= m > 1), whose solutions are known to become unbounded in finite time. To reproduce the finite-time blow-up phenomenon numerically, the temporal grid sizes are defined adaptively. We show not only that our numerical solution blows up in finite time but also that the numerical blow-up time converges to the blow-up time of the PDE. In addition, we also investigate the behavior of the numerical solutions. Our numerical solutions reproduce certain significant blow-up properties of the continuous solutions.