Abstract
The questions of approximate solution of unstable problems for evolutionary second order equations are discussed in this paper. The classical Cauchy problem for elliptic type equation is a significant example of such problem. Incorrectness of this problem (the Hadamard example) is due to instability of the solution towards small perturbations of the initial conditions. The extension problem of the solutions of well-posed elliptic problems beyond the calculation region boundary is also discussed. The stability of corresponding difference schemes is investigated by basing on general theory of ρ-stability. The principle of the regularization of three-layer difference schemes is developed for the unstable problems. It is shown that the regularized difference schemes correspond to some modification of quasi-inversion method.
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