Operator-Difference Scheme with a Factorized Operator

Abstract

In the study of difference schemes for time-dependent problems of mathematical physics, the general theory of stability (well-posedness) for operator-difference schemes is in common use. At the present time, the exact (matching necessary and sufficient) conditions for stability are obtained for a wide class of two- and three-level difference schemes considered in finite-dimensional Hilbert spaces.

The main results of the theory of stability for operator-difference schemes are obtained for problems with self-adjoint operators. In this work, we consider difference schemes for numerical solution of the Cauchy problem for first order evolution equation, where non-self-adjoint operator is represented as a product of two non-commuting self-adjoint operators. We construct unconditionally stable regularized schemes based on the solution of a grid problem with a single operator multiplier on the new time level.