We study the convection of an incompressible heat-conducting fluid saturating a long horizontal cylinder filled with a porous medium. Based on the Darcy model and equations with respect to the temperature and stream function in polar coordinates, a finite-difference scheme with a special approximation in the vicinity of pole is developed. The problem with nonuniform boundary temperature distributions is considered to detect convection occurrence. In the case of linear in height the temperature profile we derive formulas based on the zeros of Bessel functions to determine the critical values of Rayleigh numbers. The results of numerical calculation of convective regimes branching off of state of rest are presented.