The proof of stability of solutions to difference schemes is one of the central problems in the construction and analysis of numerical methods for nonstationary differential equations. Existing approaches to stability analysis, such as the maximum principle, the method of energy inequalities, the harmonic (Fourier) method, and others, provide effective results for problems with constant coefficients. However, when dealing with differential equations with variable (in space and time) coefficients, the analysis of the stability of the corresponding numerical algorithms becomes significantly more complicated, making it difficult to establish the stability of their solutions. In this paper, an attempt is made to study the stability of difference schemes based on new stability criteria for difference equations. In particular, criteria for uniform and uniformly asymptotic stability of solutions to difference equations are obtained. A discrete analogue of the well-known Wazewski inequality is developed, which makes it possible to derive sufficient conditions for uniform asymptotic stability. In addition, criteria for uniform stability of solutions to homogeneous difference equations are obtained using the second Lyapunov method. Theorems on uniform and uniformly asymptotic stability of difference equations are proved. The obtained results are applied to the study of the stability of two-layer difference schemes arising in the approximation of initial-boundary value problems for nonstationary equations with variable coefficients using the finite difference method and the finite element method. As examples, initial-boundary value problems for diffusion and convection–diffusion equations with variable coefficients are considered. The results demonstrate that the proposed criteria provide an effective tool for analyzing the stability of difference schemes constructed for a wide class of nonstationary differential equations with variable coefficients.
Utebaev et al. (Wed,) studied this question.