Abstract <p>The problem of convection in an incompressible, heat-conducting fluid that saturates a long horizontal cylinder filled with a porous medium is considered. Based on the Darcy model and the equations in cylindrical coordinates, a scheme with staggered grids, and a nonuniform distribution of nodes, as well as a special approximation near the pole, is developed. To discretize the problem in primitive variables, the integro-interpolation method is used. The discrete transition to equations in terms of the stream function and temperature is performed. To determine the critical values of the Rayleigh number, formulas based on the zeros of Bessel functions are derived. The results of calculating the critical Rayleigh numbers are presented. Experiments are performed for a nonuniform distribution of nodes. The convective regimes that branch off from mechanical equilibrium are calculated.</p>