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 domain 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 executes 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 R. Bellman’s optimal control principle) and fast Fourier transform. Mathematical software written in C/C++ has been developed. Examples of solving model problems with various directions of the heating flow of a square domain relative to the vibration vector are given.