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

Abstract

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 unbounded growth. Based on sufficient conditions for the stability of A.A. Samarskii’s two- and three-level 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 nonnegative at all nodes of the grid domain. The results obtained are applied to the analysis of the stability of difference schemes that approximate the Fisher and Klein–Gordon equations with nonlinear right-hand sides.