An effective multigrid method for solving problems of high frequency vibrational convection

The paper describes an implemented algorithm for solving the problem of vibrational convection in a rectangular area filled with an unevenly heated incompressible fluid. The mathematical model is based on the solution of the Simonenko—Zenkovskaya equations obtained by averaging the Navier—Stokes equations under the assumption that the volume of liquid performs high-frequency translational vibrations. To solve the Poisson equations, an algebraic multigrid method is implemented in combination with a highly efficient dynamic programming method (based on the optimal control principle of R. Bellman) and fast Fourier transform. Mathematical software written in C/C++ has been developed. Examples of solving model problems with different directions of the heating flow of a square region relative to the vibration vector are given.