Convergence of euler‐stokes splitting of the navier‐stokes equations

Abstract

We consider approximation by partial time steps of a smooth solution of the Navier‐Stokes equations in a smooth domain in two or three space dimensions with no‐slip boundary condition. For small k > 0, we alternate the solution for time k of the inviscid Euler equations, with tangential boundary condition, and the solution of the linear Stokes equations for time k, with the no‐slip condition imposed. We show that this approximation remains bounded in H2,p and is accurate to order k in Lp for p > ∞. The principal difficulty is that the initial state for each Stokes step has tangential velocity at the boundary generated during the Euler step, and thus does not satisfy the boundary condition for the Stokes step. The validity of such a fractional step method or splitting is an underlying principle for some computational methods. © 1994 John Wiley & Sons, Inc. Copyright © 1994 Wiley Periodicals, Inc., A Wiley Company