A difference scheme for Cauchy problem for the hyperbolic equation with self-adjoint operator

A new second order absolutely stable difference scheme is presented for Cauchy problem for second-order hyperbolic differential equations containing the operator A(t)A(t). This scheme makes use of this operator which is unbounded linear self-adjoint positive definite with domain in an arbitrary Hilbert space. The stability estimates for the solution of this difference scheme and for the first and second-order difference derivatives are established. The theoretical statements for the solution of this difference scheme are supported by the results of numerical experiments.