Main scientific results of Academician A.A. Samarskii in computational mathematics.

Significant contribution of A.A. Samarskii to the theory of iterative methods is the development of the universal alternating-triangular method, construction and justification of the iterative alternating direction method for solving boundary value problems of difference analogues of higher order of accuracy for the Poisson equation, the solution of the problem of computing the stability of the Chebyshev iteration method.

In the paper "On one economy algorithm of numerical solution of systems of differential and algebraic equations" (J. Comp. Math. and Math. Phys., Vol.4, No.3, 1964. pp.580-595), A.A. Samarskii suggested an economy (for the number of operations) differential scheme of the second order of accuracy for the solution of a system of ordinary differential equations of the first order.

In the case of usage of an explicit scheme for the Cauchy problem for the system of equations, the number of arithmetic operations is q=2n²+2n, i.e. O(n²). An implicit scheme has second order of accuracy but for it q=O(n³). A.A. Samarskii suggested to consider two-step difference scheme that contains two triangular positive matrices, the sum of which equals to the matrix of the initial system:

The calculation cost of using this scheme is q=n²+7n, i.e. at n>5 the cost for this two-step scheme is even less than that for the explicit one with the step τ. If A₁, A₂ are self-conjugate operators, this scheme is a generalization of a known algorithm of alternating directions for the two-dimensional equation of heat conductivity.  

The same scheme can be used as an iterative scheme for the solution of a system of linear algebraic equations, and all results are transferred to the case of linear operator equations.

A.A. Samarskii and E.S. Nikolaev in the paper "Selection of iterative parameters in the Richardson method" (J. Comp. Math. and Math. Phys., 1972, Vol. 12, No.4 pp.960-973)  considered the problem of computational stability of iterative method of Richardson for the solution of an operator equation of the 1st order in the Hilbert space.

This method has high convergence speed but in practice for the problems with ill-conditioned operator, it was found to be numerically unstable due to the rounding errors during calculations on computers. Before, other authors suggested the methods to minimize instability but not how to eliminate it completely. Then it was suggested how to eliminate the instability but only for the number of parameters n=2^p. A.A. Samarskii and E.S. Nikolaev generalized the method of ordering of parameters for an arbitrary n.