About convergence for finite-difference equations of incompressible fluid with boundary conditions by Woods formulas

In this paper, mathematical aspects of stability, convergence and numerical implementation of two-dimensional differential problem for incompressible fluid equations in “stream function, vorticity” variables defined on a symmetrical template of finite-difference grid studied by method of a priori estimates are considered. Approximate boundary conditions for the vorticity are chosen in the form of Woods formula. In case of a linear Stokes problem, it is shown that the numerical solution of the difference problem converges to the solution of the differential problem with second order accuracy and two algorithms of numerical implementation, for which the rates of convergence obtained, are considered. In the case of non-linear Navier-Stokes equations, estimates of the convergence of a solution of the difference problem to the solution of the differential problem, as well as estimation of the convergence of a considered iterative algorithm with the assumption that the condition is equivalent to the condition of uniqueness of nonlinear difference problem are obtained.