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Splitting and alternating direction methods. (Metody rasshchepleniya i peremennykh napravlenij). (Russian)
Moskva: Otdel Vychislitel’noj Matematiki AN SSSR. 334 p. R. 2.10 (1986).
Splitting methods are one of the fundamental tools of modern numerical analysts which are hardly to avoid at the numerical solution of many complicated boundary value problems. Some essential advances in computations and especially in the solution of gas and fluid dynamics problems are due to such methods. To be fair it should be added that in the solution of simple elliptic equations considerable initial successes were achieved by the use of alternating direction methods. In the book a wide variety of splitting algorithms are systematized and considered in different aspects. It consists of three parts devoted correspondingly to the construction of splitting methods, to problems of their investigation and to a consideration of splitting schemes for some particular problems. The author starts from a review of the basic notions and concepts of the finite-difference method like approximation, stability, convergence, Crank-Nicolson’s scheme and so on. After that the ways of construction of the main splitting algorithms are described, in particular such algorithms as splitting correspond to the components of the solution, splitting with factorization of operators, splitting in the form of the predictor-corrector method. The alternating direction method and the method of stabilizing corrections, splitting according to physical processes, and the alternating triangular method. Among the methods of investigation first of all a Fourier analysis and the method of a priori estimates are considered. The rest of the second part is devoted to the analysis of splitting algorithms based on the concept of weak approximation, to the choice and optimization of iteration parameters when splitting algorithms are used for iterative solution of algebraic systems and to an analysis of splitting and decomposition methods for variational problems. In the last part of the book splitting algorithms are considered for some specific problems like problems of thermoconductivity, hydrodynamics, oceanology and others. The list of references consists of 567 items. Results of Soviet and other authors are widely quoted. If the proof of a result is omitted, usually the reference is given where it can be found. The book can serve as a good introduction in the ideas and results of the large subfield of numerical analysis occupied by splitting methods.
Reviewer: V.Korneev