On the finite difference approximation for blow-up solutions of the porous medium equation with a source

Abstract

We consider the porous medium equation with a nonlinear source term, ut=(um)xx+uβ, x∈(0,L), t>0, whose solution blows up in finite time. Here β⩾m>1, L>0 are parameters. To approximate the solution near blow-up time and to estimate the blow-up time numerically, the concept of adaptive time meshes is introduced so as to construct a finite difference scheme whose solution also blows up in finite time. For such schemes, we show not only the convergence of the numerical solution but also the convergence of the numerical blow-up time. Moreover, the numerical blow-up sets are classified. It is interesting that although the convergence of the numerical solution is guaranteed, the numerical blow-up sets are sometimes different from that of the PDE. However, the blow-up shapes are reproduced numerically by our schemes.