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

Ashyralyev, Allaberen; Sobolevskii, Pavel E.

Discrete Dyn. Nat. Soc. 2005, No. 2, 183-213 (2005).

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.

Reviewer: Răzvan Răducanu (Iaşi)

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

Keywords:

Hilbert space; Taylor decomposition; hyperbolic equations; abstract Cauchy problem; two-step difference schemes; stability