Two new approaches for construction of the high order of accuracy difference schemes for hyperbolic differential equations

We consider the abstract Cauchy problem for differential equation of the hyperbolic type v ''(t) + Av(t) = f(t) (0 <= t <= t), v(0) = v(0), v'(0) = v(o)' in arbitrary Hilbert space H with the selfadjoint positive definite operator A. The high order of accuracy two-step difference schemes generated by an exact difference scheme or by the Taylor decomposition on the three points for the numerical solutions of this problem are presented. The stability estimates for the solutions of these difference schemes are established. In applications, the stability estimates for the solutions of the high order of accuracy difference schemes of the mixed-type boundary value problems for hyperbolic equations are obtained.