Numerical blow-up for a nonlinear heat equation

Abstract

This paper concerns the study of the numerical approximation for the following initialboundary value problem $$ \left\{ \begin{gathered} u_t - u_{xx} = f\left( u \right), x \in \left( {0,1} \right), t \in \left( {0,T} \right), \hfill \\ u\left( {0,t} \right) = 0, u_x \left( {1,t} \right) = 0, t \in \left( {0,T} \right), \hfill \\ u\left( {x,0} \right) = u_0 \left( x \right), x \in \left[ {0,1} \right], \hfill \\ \end{gathered} \right. $$ where f(s) is a positive, increasing, C1 convex function for the nonnegative values of s, f(0) > 0, \( \frac{{ds}} {{f\left( s \right)}} \) < ∞, u0 ∈ C1([0, 1]), u0(0) = 0, u′0(1) = 0. We find some conditions under which the solution of a semidiscrete form of the above problem blows up in a finite time and estimate its semidiscrete blow-up time. We also prove the convergence of the semidiscrete blow-up time to the theoretical one. A similar study has been also undertaken for a discrete form of the above problem. Finally, we give some numerical results to illustrate our analysis.