Numerical analysis of the rescaling method for parabolic problems with blow-up in finite time

Abstract

In this work, we study the numerical solution for parabolic equations whose solutions have a common property of blowing up in finite time and the equations are invariant under the following scaling transformation $$u \mapsto u_λ(x,t):= λ^{\frac{2}{p-1}}u(λx, λ^2 t).$$ For that purpose, we apply the rescaling method proposed by Berger and Kohn in 1988 to such problems. The convergence of the method is proved under some regularity assumption. Some numerical experiments are given to derive the blow-up profile verifying henceforth the theoretical results.