Main scientific results of Academician A.A. Samarskii in the theory of difference methods.
A.A. Samarskii obtained fundamental results in the theory of grid approximation of the equations of mathematical physics, in the theory of stability of difference schemes, in the theory of construction and validation of the methods of solution of grid equations.
The papers «Homogeneous difference schemes», «On convergence of difference schemes in the class of discontinuous functions» и «On one best homogeneous difference scheme» laid the foundations of the theory of homogeneous difference schemes. It is not practical to create difference schemes and programs designed only to address specific problems of a particular type, it is necessary to have difference schemes suitable for solving classes of problems defined only by specifying the type of differential equation and boundary conditions.
The homogeneous difference scheme is meant a the difference scheme, which form does not depend on the choice of a particular problem in this class, nor on the choice of the difference grid: the difference equations in all grid points must be of the same form and it is desirable that the scheme would be suitable for the equations with discontinuous coefficients.
A.A. Samarskii constructed heuristic examples and formulated common principles (in particular, the principle of full conservatism), which allowed to create homogeneous schemes. He also analyzed the properties of such schemes.
For the homogeneous difference schemes, the principle of conservatism was formulated as a required condition of convergence in the class of discontinuous coefficients. A.A. Samarskii suggested difference schemes, which are suitable for both continuous and discontinuous coefficients of the equation.
In the theory of economy schemes, A.A. Samarskii formulated the principle of full approximation. This principle was used to derive economy difference schemes for the common equations of mathematical physics in the domains of complex shape.
In the series of papers devoted to difference methods of solution of non-stationary multi-dimensional problems of mathematical physics, A.A. Samarskii extended the method of a priori estimates, which allowed to obtain the estimate of the convergence speed of difference schemes in different metrics.
Fundamental results were obtained by A.A. Samarskii in the theory of stability of difference schemes. This theory is the central part of the whole theory of difference schemes.
Traditional spectroscopic methods of stability analysis, as a rule, used assumptions about the structure of difference operators, gave difficult to use and inefficient results, and for non-selfadjoint operators, these methods gave only the necessary conditions for stability.
A. Samarskii treats difference scheme as an operator equation or operator-difference equation in an abstract Hilbert space and explores stability as an intrinsic property of the scheme, which is independent of the approximation and connection of the scheme with any differential equation.
A significant achievement of A.A. Samarskii in the stability theory was to find the necessary and sufficient conditions for the stability of difference schemes of a very general form in terms of inequalities for the operators from the differential equation. Let us write in the simplest form two relevant results about two-level scheme of the form
Theorem 1. Let A and B do not depend on k,
Then the condition
is necessary and sufficient for the stability of the scheme (1), i.e. for the equality
Theorem 2. Let
Then the condition
where ρ - is an arbitrary number, is necessary and sufficient for the stability of the scheme (1) and for the equality
Similar results were obtained by A.A. Samarskii for more general schemes and for the case where A and B depend on k. These results are one of the fundamental achievements of the modern theory of the stability of difference schemes.
Assuming in (1) that
and choosing the operator R to fulfill the corresponding inequalities, one can "regularize", i.e. to make many schemes stable. Such a regularization is particularly useful for the three-level schemes, where it allows to keep the second-order approximation in τ.
In recent years, A.A. Samarskii, developing his theory, obtained outstanding results related to the difference schemes for the unstable problems of mathematical physics, he transferred a number of results of the stability theory of difference schemes to the projection methods, including the finite element method.