A note on the difference schemes for hyperbolic equations

The initial value problem for hyperbolic equations d 2 u ( t )/ d t 2 + A u ( t ) = f ( t )(0 ≤ t ≤ 1), u (0) = φ , u ′ (0) = ψ , 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.