USING OPERATOR INEQUALITIES IN STABILITY RESEARCH OF DIFFERENCE SCHEMES FOR NONLINEAR BOUNDARY PROBLEMS WITH NONLINEARITY OF UNLIMITED GROWTH

The article develops the theory of stability of linear operator schemes for operator inequalities and nonlinear nonstationary initial-boundary value problems of mathematical physics with nonlinearities of unlimited growth. Based on sufficient conditions for the stability of A.A. Samarsky’s two-layer and three-layer difference schemes, the corresponding a priori estimates for operator inequalities are obtained under the condition of the criticality of the difference schemes under consideration, i.e. when the difference solution and its first time derivative are non-negative at all nodes of the grid region. The results obtained are applied to the analysis of the stability of difference schemes that approximate the Fisher and Klein-Gordon equation with a nonlinear right-hand side.