THE EULER METHOD IN THE BLOW-UP NUMERICAL SOLUTIONS FOR A REACTION-DIFFUSION PROBLEMS WITH BOUNDARY CONDITIONS

This paper concerns the study of the numerical approximation for the following initial-boundary value problem

$$\left\{\begin{array}{ll}\hbox{$u_t(x,t)-u_{xx}(x,t)= \gamma e^{u(a,t)},\quad x\in(0,1),\quad t\in(0,T)$,} \\\hbox{$u(0,t)=0, \quad u_x(1,t)=0,\quad t\in(0,T)$,} \\ \hbox{$u(x,0)=u_{0}(x)\geq0,\quad x\in[0,1]$,} \\\end{array}\right.$$

where $u_0\in C^1([0,1]),$ $u_0(0)=0,$ $u'_0(1)=0.$ $a\in(0,1)$, $\gamma $ is a positif parameter. 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 study the asymptotic behavior of a semi-discrete numerical approximation. 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. Also obtaining results on the convergence of the numerical blow-up times to the theoretical limit when the mesh parameter is small enough.