A note on the difference schemes for hyperbolic equations

Abstract

The initial value problem for hyperbolic equations $d^{2}u\left(t\right)/d{t}^2+Au\left(t\right)=f\left(t\right)\left(0\le t\le 1\right),u\left(0\right)=\varphi ,u^{\prime }\left(0\right)=\psi$, in a Hilbert space $H$ is considered. The first and second order accuracy difference schemes generated by the integer power of $A$ approximately solving this initial value problem are presented. The stability estimates for the solution of these difference schemes are obtained.