Adaptive algorithms for convection-diffusion-reaction equations of quenching type (Q2724369)

The authors consider the nonlinear convection-diffusion-reaction initial boundary value problem NEWLINE\[NEWLINEu_t -\sigma^2u_{xx}-p(x)u_x = f(u),\quad (x,t)\in (0,1)\times (t_0, T);\tag{1}NEWLINE\]NEWLINE NEWLINE\[NEWLINEu(0,t) = u(1,t) = 0,\quad t\in(t_0,T);\quad u(x,t_0)=0,\quad x\in (0,1)\tag{1}NEWLINE\]NEWLINE modelling quenching phenomena, where \(0 < \sigma < 1\) and \(p(x)\) is nonnegative and bounded from above. For some special case, it is known that the quenching phenomenon appears at some finite time \(T_\sigma > t_0\) if \(\sigma\) is less than some critical value \(\sigma^*\).NEWLINENEWLINENEWLINEThe authors propose some adaptive finite difference scheme that computes some approximate solution to (1)--(2) along with an approximate prediction of the critical value \(\sigma^*\) and quenching time \(T_\sigma\). The numerical results presented for the benchmark problem (1)--(2) with \(p = 0\), \(f(u)=1/(1-u)^\theta\) and various \(\theta > 0\) nicely confirm the analysis provided in the paper.