We first consider the semi-linear heat equation u_t=u_{xx}+f(u) whose solution blows up in finite time. Then we consider a finite difference analogue with adaptive time meshes so that the numerical solutions blow up in finite time. The convergence of the blow-up time was also proved. It is interesting that although the convergence is guaranteed, the asymptotic behavior of the numerical solution is different from the solution of the PDE.
Next, we consider the semi-linear wave equation u_{tt}=u_{xx}+f(u) and its finite difference analogue. Several significant results have been derived, however, it seems that the stability of a 3-level finite difference scheme is not well-studied. The difficulty and our recent results will be reported.
Next, we consider the semi-linear wave equation u_{tt}=u_{xx}+f(u) and its finite difference analogue. Several significant results have been derived, however, it seems that the stability of a 3-level finite difference scheme is not well-studied. The difficulty and our recent results will be reported.