On the computation for blow-up solutions of the nonlinear wave equation

Abstract

We consider in this paper the 1-dim nonlinear wave equation $\frac{\partial ^2 u}{\partial t^2}(t,x)=\frac{\partial ^2 u}{\partial x^2}(t,x)+|u|^{1+\alpha }(t,x)\;(\alpha >0)$ and its finite difference analogue. It is known that the solution of the current equation may become unbounded in finite time, a phenomenon which is known as blow-up. Moreover, since the nonlinear wave equation enjoys finite speed of propagation, even if the solution has become unbounded at certain points, the solution continues to exist, blows up at later times and forms the so-called blow-up curve. Up to the present, however, numerical approximation for blow-up problems gave only the convergence up to the blow-up time of $\Vert u(t,\cdot )\Vert _\infty$. In fact, adaptive time mesh strategy was used to numerically reproduce the phenomenon of finite-time blow-up. Nevertheless, computation using such schemes can not be extended beyond the blow-up time of $\Vert u(t,\cdot )\Vert _\infty$. This is a fatal problem when computing blow-up solutions for the nonlinear wave equation. As a consequence, we reconsider a finite difference scheme whose temporal grid size is given uniformly and propose an algorithm with rigorous convergence analysis to numerically reconstruct the blow-up curve for the nonlinear wave equation.