Studying convergence of an iterative algorithm for numerically solving the thermal convection problems in the variables “stream function-vorticity”

Abstract

Using the method of a priori estimates, we studied the operator-difference equations that approximate the differential problems of heat convection for an incompressible liquid in the variables “stream function-vorticity,” and consider the issues of convergence of the iterative schemes for solving the equations. The boundary values for the vorticity are chosen in the form of Thom’s formulas. We give a boundedness estimate and a condition for uniqueness of the solution of the finite-difference problem. Using an auxiliary function of vorticity, we transform the finite-difference equations under consideration to the relations with homogeneous boundary conditions. Some implicit iterative algorithms are proposed for numerically solving the finite-difference equations; the estimates for the rate of convergence are obtained for these algorithms provided that the conditions equivalent to the condition of uniqueness are satisfied. The behavior of iterations is analyzed in the case of the Stokes linear problem. To illustrate the advantages of the considered iterative algorithms, a problem in a closed domain with side heating is considered. The calculations are carried out for the iterative algorithm of the type of variable directions.