Stability of three-layer differential-difference schemes with weights in the space of summable functions with supports in a network-like domain

V. N. Hoang ; V. V. Provotorov

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 23 (2023) no. 3, pp. 357-369

Résumé

The work is a natural continuation of the authors' earlier studies in the analysis of the conditions for the weak solvability of one-dimensional initial-boundary value problems with a space variable changing on a graph (network) in the direction of increasing the dimension $n$ ($n>1$) of the network-like domain of change of this variable. The first results in this direction (for $n = 3$) were obtained by one of the authors for the linearized Navier – Stokes system, later for a much more complex nonlinear Navier – Stokes system. The analysis was carried out in the classical way, using a priori estimates for the norms of weak solutions in Sobolev spaces of functions. In this study (for arbitrary $n>1$) another approach is proposed to obtain conditions for the weak solvability of linear initial-boundary value problems reduction of the original problem to a differential-difference system, the idea of which goes back to E. Rothe's method of semi-discretization of the initial-boundary value problem by temporary variable. A differential-difference system of equations with weighted parameters and its corresponding three-layer differential-difference scheme (a set of schemes) are considered. The resulting system is an analog of the initial-boundary value problem for a parabolic type equation with a space variable changing in a network-like domain of an n-dimensional Euclidean space. The main aim is to establish a domain of the range of weight parameters that guarantees the stability of the differential-difference scheme (continuity by the initial data of the problem), to obtain estimates for the operator norms of the weak solutions of the scheme, to construct a sequence of solutions for a differential-difference system that is weakly compact in its state space. The latter is an important element when using numerical methods of analysis of a wide class of applied multidimensional problems and constructing computational algorithms for finding approximations to their solutions. The results are applicable in applied optimization problems arising from modeling network processes of continuum transport with the help of the formalisms of differential-difference systems.