  In approximating solutions of nonstationary problems, various approaches are
used to compute the solution at a new time level from a number of simpler
(sub-)problems. Among these approaches are splitting methods. Standard
splitting schemes are based on one or another additive splitting of the
operator into "simpler" operators that are more convenient/easier for the
computer implementation and use inhomogeneous (explicitly-implicit) time
approximations. In this paper, a new class of splitting schemes is proposed
that is characterized by an additive representation of the solution instead of
the operator corresponding to the problem (called problem operator). A specific
feature of the proposed splitting is that the resulting coupled equations for
individual solution components consist of the time derivatives of the solution
components. The proposed approaches are motivated by various applications,
including multiscale methods, domain decomposition, and so on, where spatially
local problems are solved and used to compute the solution. Unconditionally
stable splitting schemes are constructed for a first-order evolution equation,
which is considered in a finite-dimensional Hilbert space. In our splitting
algorithms, we consider the decomposition of both the main operator of the
system and the operator at the time derivative. Our goal is to provide a
general framework that combines temporal splitting algorithms and spatial
decomposition and its analysis. Applications of the framework will be studied
separately.
