An adaptive numerical method for the wave equation with a nonlinear boundary condition

Abstract

We develop an efficient numerical method for studying the existence and non-existence of global solutions to the initial-boundary value problem {gather*} u_{tt}=u_{xx}quad 00, -u_{x}(0,t)=h(u(0,t)) quad t>0, u(x,0)=f(x),quad u_{t}(x,0)=g(x) quad 0<x<infty. end{gather*} The results by this numerical method corroborate the theory presented in cite{AD}. Furthermore, they suggest that blow-up can occur for more general nonlinearities $h(u)$ with weaker conditions on the initial data $f$ and $g$.

Keywords

• Time-adaptive numerical method
• blow-up time
• blow-up rate.