Alternating Triangular Schemes for Second-Order Evolution Equations

Abstract

Schemes of the Samarskii alternating triangular method are based on splitting the problem operator into two operators that are conjugate to each other. When the Cauchy problem for a first-order evolution equation is solved approximately, this makes it possible to construct unconditionally stable two-component factorized splitting schemes. Explicit schemes are constructed for parabolic problems based on the alternating triangular method. The approximation properties can be improved by using three-level schemes. The main possibilities are indicated for constructing alternating triangular schemes for second-order evolution equations. New schemes are constructed based on the regularization of the standard alternating triangular schemes. The features of constructing alternating triangular schemes are pointed out for problems with many operator terms and for second-order evolution equations involving operator terms for the first time derivative. The study is based on the general stability (well-posedness) theory for operator-difference schemes.