Stability and Regularization of Three-Level Difference Schemes with Unbounded Operator Coefficients in Banach Spaces

Abstract

The problem of stability of difference schemes for second-order evolution problems is considered. Difference schemes are treated as abstract Cauchy problems for difference equations with operator coefficients in a Banach or Hilbert space. To construct stable difference schemes the regularization principle is employed, i.e., one starts from any simple scheme (possibly unstable) and derives absolutely stable schemes by perturbing the operator coefficients. The main result of this paper is the following: for the first time sufficient conditions are pointed out under which an unstable three-level difference scheme with unbounded operator coefficients in a Banach space can be regularized to a stable scheme. The principal stability condition is the strong P-positivity of the unbounded operator coefficients.