Study of an implicit iterative scheme for a numerical solution of stationary heat convection equations based on the idea of "weak compressibility"

This paper investigates a class of fully implicit iterative schemes for the numerical solution of stationary thermal convection equations formulated in primitive variables (velocity-pressure-temperature). The proposed approach is based on the concept of weak compressibility, which relaxes the incompressibility constraint within a pseudotime-iterative framework. From an algorithmic perspective, the resulting scheme can be interpreted as a fully implicit realization within the broader class of projection and pressure-correction methods widely used in computational fluid dynamics. A rigorous theoretical analysis of the discrete problem is presented. A priori estimates guaranteeing the stability of the numerical solution are derived, and a uniqueness condition for the discrete convection problem is established. For the linear Stokes case, the convergence of the iterative algorithm is proven, and it is shown that the iteration process converges with a geometric rate whose constants are independent of the spatial grid step. Numerical experiments are performed for the classical natural convection benchmark problem in a square cavity. The results demonstrate grid convergence of the numerical solution, confirm the geometric convergence rate predicted by the theory, and illustrate the influence of the Rayleigh number and pseudotime-parameters on the behavior of the algorithm. The obtained results provide a rigorous mathematical foundation for implicit iterative algorithms closely related to projection-type methods and demonstrate their practical applicability to numerical simulation of convective flows.