Theory of stability and regularization of difference schemes and its application to ill-posed problems of mathematical physics

Abstract

Solving numerically evolutionary problems of mathematical physics, the problem of stability of difference schemes with respect to initial and boundary data is of great importance. Principal results of the general theory of stability for two- and three-level difference schemes written in the canonical form are presented in this paper. This theory has been developed by the author in 1967. Difference schemes for unsteady problems of mathematical physics are treated in the theory as abstract Cauchy problems for operator-difference problems in Hilbert spaces. The necessary and sufficient conditions for stability in different norms are also formulated for a wide class of difference schemes. To construct stable difference schemes, a general methodological approach based on the regularization principle is employed. In this approach, we can start from any simple scheme (even unstable) as the initial one. By perturbing the initial scheme operators and taking into consideration the stability conditions, absolutely stable schemes are derived. Such an approach is used to obtain the difference schemes for approximate solution of the ill-posed problems for evolutionary equations.