Solution of the Boltzmann equation involves many difficulties mainly associated with the nonlinear and nonlocal multidimensional nature of the collision term. To overcome them, two different strategies are commonly used: 1) highly efficient numerical algorithms and 2) simplified collision operators capable of reproducing the desired properties of the Boltzmann dynamics. In this work, two corresponding approaches proposed many years ago are considered and compared for some numerical examples. The Tcheremissine regularization method provides a flexible framework for designing conservative discrete-velocity methods. The Shakhov relaxation model is able to mimic the viscosity and thermal conductivity preserving the computational simplicity. The obtained results are in very good agreement, but, due to uncontrollable inaccuracy of relaxation models, some subtle discrepancies arise in regions where the velocity distribution function is far from equilibrium.