On a new class of additive (splitting) operator-difference schemes

Many applied time-dependent problems are characterized by an additive representation of the problem operator. Additive schemes are constructed using such a splitting and associated with the transition to a new time level on the basis of the solution of more simple problems for the individual operators in the additive decomposition. We consider a new class of additive schemes for problems with additive representation of the operator at the time derivative. In this paper we construct and study the vector operatordifference schemes, which are characterized by a transition from one initial the evolution equation to a system of such equations. Introduction For the approximate solution of multidimensional unsteady problems of mathematical physics there are widely used different classes of additive schemes (splitting schemes) [5,8,17]. Beginning with the pioneering works [2,6] the most simple way to construct additive schemes is in the splitting of the problem operator on the sum of two operators with a more simple structure — alternating direction methods, factorized schemes, predictor-corrector schemes etc. [12]. In the more general case of multicomponent splitting, classes of unconditionally stable operator-difference schemes are based on the concept of summarized approximation. In this way, we can construct the classic locally one-dimensional schemes (componentwise splitting schemes) [5, 8], additively-averaged locally onedimensional schemes [3, 12]. A new class of unconditionally stable schemes — vector additive schemes (multicomponent alternating direction method schemes) is actively developed (see, eg, [1,14]). They belong to a class of full approximation schemes — each intermediate problem approximates the original one. The most simple additive full approximation schemes are based on the principle of regularization of operator-difference schemes. Improving the quality of operator-difference schemes is achieved using additive or multiplicative perturbations of operators of the scheme [7]. Regularized additive schemes for evolutionary equations of the first and second order are constructed for equations as well as systems of equations [13, 15]. Both the standard schemes of splitting with respect to separate directions (locally-onedimensional schemes), splitting with respect to physical processes and regionally-additive schemes based on domain decomposition for constructing parallel algorithms for transient problems of mathematical physics [4, 10, 16]. 2000 Mathematics Subject Classification. Primary 65N06, 65M06.