Using Operator Inequalities in Studying the Stability of Difference Schemes for Nonlinear Boundary Value Problems with Nonlinearities of Unbounded Growth

The article develops the theory of stability of linear operator schemes for operatorinequalities and nonlinear nonstationary initial–boundary value problems of mathematical physicswith nonlinearities of unbounded growth. Based on sufficient conditions for the stability ofA.A. Samarskii’s two- and three-level difference schemes, the corresponding a priori estimates foroperator inequalities are obtained under the condition of the criticality of the difference schemesunder consideration, i.e., when the difference solution and its first time derivative are nonnegativeat all nodes of the grid domain. The results obtained are applied to the analysis of the stability ofdifference schemes that approximate the Fisher and Klein–Gordon equations with nonlinearright-hand sides.