Stable Boundary Conditions and Difference Schemes for Navier–Stokes Equations

Abstract

The Navier–Stokes equations at high Reynolds numbers can be viewed as an incompletely elliptic perturbation of the Euler equations. By using the entropy function for the Euler equations as a measure of “energy” for the Navier–Stokes equations we are able to obtain nonlinear “energy” estimates for the mixed initial boundary value problem. These estimates are used to derive boundary conditions that guarantee $L^2 $ boundedness even when the Reynolds number tends to infinity. Finally, we propose a new difference scheme for modeling the Navier–Stokes equations in multidimensions for which we are able to obtain discrete energy estimates exactly analogous to those we obtained for the differential equation.