Stability of a second order of accuracy difference scheme for hyperbolic equation in a Hilbert space (Q2478395)

The authors consider an initial-value problem for the hyperbolic equation \[ \begin{aligned} \frac{d^{2}u(t)}{dt^{2}}+A(t)u(t)&=f(t) \quad (0\leq t\leq T),\\ u(0)&=\varphi,\\ u'(0)&=\psi\end{aligned} \] in a Hilbert space \(H\) with the self-adjoint positive definite operator \(A(t)\). A second-order accurate difference scheme for solving this initial-value problem is proposed. The stability estimates for the solution of this difference scheme are established.