  Domain decomposition methods are essential in solving applied problems on
parallel computer systems. For boundary value problems for evolutionary
equations the implicit schemes are in common use to solve problems at a new
time level employing iterative methods of domain decomposition. An alternative
approach is based on constructing iteration-free methods based on special
schemes of splitting into subdomains. Such regionally-additive schemes are
constructed using the general theory of additive operator-difference schemes.
There are employed the analogues of classical schemes of alternating direction
method, locally one-dimensional schemes, factorization methods, vector and
regularized additive schemes. The main results were obtained here for
time-dependent problems with self-adjoint elliptic operators of second order.
  The paper discusses the Cauchy problem for the first order evolutionary
equations with a nonnegative not self-adjoint operator in a finite-dimensional
Hilbert space. Based on the partition of unit, we have constructed the
operators of decomposition which preserve nonnegativity for the individual
operator terms of splitting. Unconditionally stable additive schemes of domain
decomposition were constructed using the regularization principle for
operator-difference schemes. Vector additive schemes were considered, too. The
results of our work are illustrated by a model problem for the two-dimensional
parabolic equation.
