On error estimates for schemes of the projection-difference method for hyperbolic equations

We study the convergence of a three-level scheme of the projection-difference method for an abstract quasi-linear hyperbolic equation. We establish asymptotic energy estimates for the error. The order of these estimates is unimprovable. A preliminary result on the conditional stability of the scheme ($W$-stability in the sense of the definition formulated in the paper) forms the basis of our derivation of the estimates. We illustrate the use of our general results by an example of a scheme with finite element space discretization applied to the first initial boundary-value problem for a second-order hyperbolic equation. We also note the possibility of application of our general results in the case when the space discretization is realized by the Galerkin method in the form of Mikhlin.