Differential-difference method to solve problems of hydrodynamics

Mathematical modelling of stationary processes of heat transfer, the problem of the current function in fluid mechanics and a number of other objects leads to the Poisson equation. The solution of multidimensional Poisson equations using various approximation schemes is usually brought to sweep methods or to problems of establishment, for which fictitious time is introduced. When solving two-dimensional problems of hydromechanics, the main part of the required machine time is spent on solving the equation of the current function, which has the form of the Poisson equation. When solving this problem, you can use the differential-difference method and with this eliminate the stage of matching the results of the sweep along different coordinate axes. In this paper, an algorithm based on this method, is proposed. The core of the differential-difference method is the diagonalization of the matrix equation using eigenvalues and vectors of the main three-diagonal transition matrix from differential to finite-difference operators. Twice applying the method of lines for a two-dimensional problem, solutions of the problem for new variables are obtained, which are linear combinations of the original grid unknowns. A formula for the reverse transition to the original unknowns is presented. The efficiency and accuracy of the algorithm for solving elliptic equations with stationary and non-stationary right-hand sides are proved using a number of test problems as examples.