## Two new approaches for construction of the high order of accuracy difference schemes for hyperbolic differential equations.(English)Zbl 1094.65077

The authors consider the abstract Cauchy problem for a differential equation of hyperbolic type $$v^{\prime \prime }(t)+Av(t)=f(t)$$ $$(0\leq t\leq T),$$ $$v(0)=v_{0},$$ $$v^{\prime }(0)=v_{0}^{\prime }$$ in an arbitrary Hilbert space $$H$$ with the selfadjoint positive definite operator $$A$$. 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. Stability estimates for the solutions of these schemes are established. In applications, the stability estimates for the solutions of high order of accuracy difference schemes of mixed-type value problems for hyperbolic equations are obtained.

### MSC:

 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 34G10 Linear differential equations in abstract spaces 35L15 Initial value problems for second-order hyperbolic equations 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65L12 Finite difference and finite volume methods for ordinary differential equations 65L20 Stability and convergence of numerical methods for ordinary differential equations
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