About one splitting scheme for the nonlinear problem of thermal convection

摘要

The paper is devoted to the questions of mathematical modeling of heating processes in hydrodynamics, namely to development and to mathematical justification of numerical algorithms for solving three-dimensional equations of free convection in natural variables. The purpose is to research implicit iterative schemes for the numerical solution of Boussinesq-type fixed (stationary) equations. The research uses mathematical modeling, mathematical programming, the Visual Fortran programming language, and the Axum 7.0 graphics program. Computational mathematics and functional analysis methods are used for the mathematical argumentation of iterative algorithms. The questions of convergence and estimate of a degree of convergence of one nonlinear splitting algorithm are considered, made for the difference analogues of the system of free convection steady-state equations in variables "velocity vector and pressure", written to shifted grids with symmetric approximation. The implicit iterative splitting algorithms for the difference analogues of the system of free convection steady-state equations in variables "velocity vector and pressure" are considered, written to shifted grids with symmetric approximation. The problems of stability of the difference problems according to the initial data and the right member, convergence and estimate of the linear algorithm degree of convergence were studied. The results of this research can be useful in studies on difference schemes for hydrodynamic equations; they can also be used to further develop the theory of numerical solution of mathematical physics problems. The research results may be used in information system development for the automation of heat-aggregation exchange problem solving and as a teaching material for students learning mathematics, mechanics, and IT technologies.