Stability and regularization of three-level difference schemes with unbounded operator coefficients in Banach spaces. (English) Zbl 1002.65101

Samarskii, Alexander A.; Gavrilyuk, Ivan P.; Makarov, Vladimir L.

SIAM J. Numer. Anal. 39, No. 2, 708-723 (2001).

The problem of stability of difference schemes for second-order evolution equations is considered. The difference schemes are treated as abstract Cauchy problems for difference equations with operator coefficients in a Banach or Hilbert space. The regularization principle is employed to construct stable difference schemes. It starts from any simple scheme and derives absolutely stable schemes by perturbing the operator coefficients.
The aim of this paper is to obtain stability results for regularized three-level difference schemes with unbounded operator coefficients in a Banach space. For example this class of schemes arises when approximating second-order evolution differential equations.

Reviewer: L.Hącia (Poznań)

MSC:

65M12	Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M06	Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M20	Method of lines for initial value and initial-boundary value problems involving PDEs
65J10	Numerical solutions to equations with linear operators
34G10	Linear differential equations in abstract spaces
35K90	Abstract parabolic equations

Keywords:

three-level difference schemes; unbounded operator coefficients; strongly P-positive operators; \(\rho\)-stability; abstract Cauchy problems; regularization; Banach space; second-order evolution differential equations