One method for the numerical solution of the Cauchy problem for the Navier-Stokes equations and Fourier series of the curl operator

摘要

The Cauchy problem for the Navier–Stokes equations with a periodicity condition in spatial variables is considered in a threedimensional space that uniformly rotates about the vertical axis. The study of this problem is based on Fourier series expansions of the given and unknown vector functions in terms of periodic eigenfunctions of the curl and Stokes operators.By applying the Galerkin method, the problem is reduced to a Cauchy problem for a system of ordinary differential equations, which has a simple explicit form in the basis under consideration. Its linear part is diagonal, while the nonlinear part in each equation is a quadratic form of the unknown functions, whose coefficients are calculated in terms of the scalar products of the curl basis vectors. A new way of numerically solving the problem is opened up, which is implemented in this work. While developing this method,which is referred to as spectral-analytical, we implemented software codes for computing the Fourier series expansion coefficients of the given vector function as expanded in terms of the eigenfunctions of the curl operator, for recovering Galerkin systems, for the numerical solution of the Cauchy problem for these systems, and others.