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Domain decomposition methods are essential in solving applied problems onparallel computer systems. For boundary value problems for evolutionaryequations the implicit schemes are in common use to solve problems at a newtime level employing iterative methods of domain decomposition. An alternativeapproach is based on constructing iteration-free methods based on specialschemes of splitting into subdomains. Such regionally-additive schemes areconstructed using the general theory of additive operator-difference schemes.There are employed the analogues of classical schemes of alternating directionmethod, locally one-dimensional schemes, factorization methods, vector andregularized additive schemes. The main results were obtained here fortime-dependent problems with self-adjoint elliptic operators of second order.  The paper discusses the Cauchy problem for the first order evolutionaryequations with a nonnegative not self-adjoint operator in a finite-dimensionalHilbert space. Based on the partition of unit, we have constructed theoperators of decomposition which preserve nonnegativity for the individualoperator terms of splitting. Unconditionally stable additive schemes of domaindecomposition were constructed using the regularization principle foroperator-difference schemes. Vector additive schemes were considered, too. Theresults of our work are illustrated by a model problem for the two-dimensionalparabolic equation.

Domain decomposition schemes for evolutionary equations of first order with not self-adjoint operators