Net schemes for the navier—stokes equations associated with projection

Abstract

DIFFERENCE schemes of the projection type for the Navier Stokes equations are considered. In the construction of schemes of the alternating directions type the continuity equation is replaced by a regularized equation. For all the difference schemes considered the convergence of the approximate to the exact solution is proved and estimates of the rate of convergence are given. In [1–4] schemes were considered which reduce at each time step the solution of a Navier- Stokes equation to the solution of quasilinear parabolic equations for the velocity components and to a subsequent projection of the velocity vector obtained in this way onto a space of solenoid functions. This requires a Poisson equation to be solved at each time step. In this paper schemes are proposed combining the projection with the method of alternating directions. Their construction uses methods of regularizing the continuity equation (see [5–7]) and the limitation of non-linearity (see [8, 9]). For all the schemes considered in this paper convergence is proved and estimates are given of the rate of convergence without restrictions on the nature of the dependence of r on h and v, where T is the time step, h is the step along each of the spatial variables, and v is the value of the kinematic viscosity.