Differential Equations, Vol. 37, No. 12, 2001, pp. 1751–1760. Translated from Differentsial’nye Uravneniya, Vol. 37, No. 12, 2001, pp. 1664–1673. Original Russian Text Copyright c  2001 by Abrashin, Egorov, Zhadaeva. NUMERICAL METHODS. FINITE-DIFFERENCE EQUATIONS On a Class of Additive Iterative Methods V. N. Abrashin, A. A. Egorov, and N. G. Zhadaeva Institute for Mathematics, National Academy of Sciences, Minsk, Belarus Belarus State University, Minsk, Belarus Received May 28, 2001 INTRODUCTION In the solution of complicated problems of mathematical physics by numerical methods, great attention is paid to the construction and investigation of efficient algorithms [1–4]. These methods are based on an additive representation of the space operator as the sum of simpler operators, which permits one to pass from a complicated problem to a chain of elementary problems, which admit either sequential or parallel solution. Such methods include classical alternating direction methods, locally one-dimensional finite-difference schemes, and others. Decomposition schemes with respect to separate directions in multidimensional problems, subdomain decomposition schemes [4, 5], and decomposition into physical processes can also be viewed as additive methods. In view of the properties of additive schemes, they occupy an intermediate position between explicit and implicit finite-difference schemes: the amount of computations is close to that of explicit schemes, and from the viewpoint of the stability, they are close to absolutely stable implicit methods. Note that efficient methods are quite completely investigated, and the domain of their appli- cations is very wide. However, the investigation of additive algorithms faces new notions, which are not specific to ordinary implicit finite-difference schemes. First of all, such notions include the total (weak) consistency introduced in [2, p. 410]. Decomposition methods do not possess complete consistency and are consistent only in the total sense, which is a source of an addi- tional error in the algorithm and substantially deteriorates the asymptotic properties of implicit finite-difference schemes [6]. The fact that the algorithm becomes conditionally stable for decompo- sition into three or more components is a serious disadvantage of the alternating direction method. The pairwise commutativity of the space operators is necessary for the stability of classical factor- ization schemes [2, 7], which also substantially restricts the applications of this class of additive finite-difference schemes. Additive methods are widely used for the solution of multidimensional stationary problems. Alternating direction schemes and factorization methods are optimal stabilization methods for the corresponding nonstationary equations. From the viewpoint of two-layer finite-difference schemes, in many cases, an efficient exit to a stationary mode provides the validity not only of solution conservation laws, related to the conservativeness of the algorithm, but also of the laws governing the time evolution of the solution, which is related to asymptotic characteristics of finite-difference methods [8, p. 207]. Multicomponent alternating direction methods (MADM) suggested in [9, 10] were considered for both sequential and parallel organization of computations. These methods can be viewed as schemes of complete consistency and are absolutely stable for a multicomponent additive decom- position of the original operator even if the pairwise commutativity of decomposition components is not required. Note that multicomponent alternating direction methods have good asymptotic properties, which permits one to use them effectively in the construction of iterative methods for stationary problems [11–14]. In the present paper, we consider iterative multicomponent alternating direction methods of sequential type for various ways of the multicomponent decomposition. 1. STATEMENT OF THE PROBLEM AND THE CONSTRUCTION OF MADM ALGORITHMS Consider the first-order operator equation Ay = f, (1.1) 0012-2661/01/3712-1751$25.00 c  2001 MAIK “Nauka/Interperiodica”