1LIMSI (CNRS), Orsay, France
2Politecuico di Milano, Piazza Leonardo da Vinci, Milano, Italia
{\noindent \bf 1. Introduction}
Achieving high order time-accuracy in the solution of the incompressible Navier-Stokes equations by means of fractional step projection methods is a nontrivial task. In fact, a basic feature of projection methods is the uncoupling of the advection-diffusion mechanism from the incompressibility condition, the consequence of this uncoupling being the introduction of a {\it time-splitting error } that is an obstacle to develop high order schemes. In particular, the accuracy of the nonincremental projection method introduced by Chorin [1] and Temam [8] is limited by an irreducible time-splitting error of ${\mathcal O}(\Delta t)$ [7] that prevents second-order accuracy.
This limitation is not present in the incremental version of the projection method, also known as ``pressure correction method,'' originally proposed by Goda [3] in a finite difference context. The finite element counterpart of this scheme based on a first-order Euler time-discretization has been analyzed thoroughly in [4] and employed successfully in [6]. The time-splitting error of such an incremental method has been numerically shown to be of ${\mathcal O}((\Delta t)^2)$ in [6]. Furthermore, this result has been recently proved in [5], where a new $(\Delta t)^2$-accurate true projection scheme based on the three-level Backward Difference Formula has been proposed, for the first time.
The aim of this talk is to report on the theoretical results of [5], to illustrate a three-dimensional finite-element implementation of the BDF scheme, and to demonstrate its second-order accuracy by means of numerical tests. In particular we shall emphasize some non standard aspects of projections methods.
{\medskip \noindent \bf 2. The numerical scheme }
The three-level BDF has been chosen to benefit from better stability properties than those of the commonly used Crank-Nicolson scheme, which is known to be marginally stable. The nonlinear term, written in skew form, has been approximated semi-implicitly to second-order accuracy using linear extrapolation in time of the advection velocity. A judicious choice of coefficients in the projection equation leads to an overall unconditional stability, such that, when mixed finite elements are used, no {\it ad hoc} stabilization technique is needed.
The present method has been implemented using mixed $P_1-P_2$ finite elements. The linear systems of equations arising from the discretization of the two half-steps of the method are solved by preconditioned GMRES provided in the SparsKit Package developed by Yousef Saad. The preconditioning consists in an incomplete LU decomposition.
{\medskip \noindent \bf 3. Numerical results and conclusions }
To verify the theoretical ${\mathcal O}((\Delta t)^2)$ accuracy, the BDF scheme has been tested with the following analytical solution in the unit square $[0, 1]^2: u_x = -\cos x \sin y \sin(2t)$, $u_y \sin x \cos y \sin(2t)$ and $p =-\frac{1}{4}[\cos(2x) + \cos(2y)] \times [\sin(2t)]^2$, for Reynolds number $Re=100$, on a mesh of $2 \times 402 P_1-P_2$ triangles. Figure 1 shows the maximum value in time, over $0 \le t \le 1.5$, of the error in the $L^2$ norm for the pressure and the error in the $H^1$ and $L^2$ norms for the velocity. The different saturations of the errors as $\Delta t \rightarrow 0$ are due to the spatial discretization error, which is of order $h^2$ for the velocity (resp. pressure) in the $H^1$ (resp. $L^2$) norm, while it is of order $h^3$ for the velocity in the $L^2$ norm.
The capability of the BDF-$(\Delta t)^2$ projection method to solve large-scale 3D problems has been assessed on the 3D driven cavity: $ (|x| \le 0.5) \times (|y|\le 0.5) \times (|z| \le 1.5)$ at $Re=3200$ in the time interval $0 \le t \le 200$ (benchmark problem documented in [2]). A nonuniform mesh of $6 \times 202 \times 29$ $P_1-P_2$ tetrahedra for the half cavity has been used in the test presented herein (87579 $P2$ nodes). The symmetry with respect to the plane $z = 0$ is assumed. In Figure 2 we give two representations of the velocity field at time $t=50$ on the plane $x=0.13$ and at time $t = 100$ on the plane $x = 0.18$. In both cases, the integral curves of the projected velocity field are shown. The qualitative comparison with analogous computations reported in [2] is satisfactory. In fact, our results show features that are very similar to those available in [2], in terms of the general pattern of the main vortices, number and location of the secondary vortices, and flow unsteadiness.
\begin{center} \epsfig{ figure=il-q1.eps,width=5.5cm } \end{center} Figure 1: Convergence tests for the second-order BDF projection method, with mixed $P_1-P_2$ interpolation. Analytical test problem for $R = 100$; finite element mesh of $2 \times 40^2$ $P_2$-triangles.
\begin{center} \epsfig{ figure=il-q2.eps,width=7.5cm } \end{center} Figure 2: Three-dimensional driven cavity problem for $Re = 3200$. (Up) velocity field at $t = 50$ projected on plane $x = 0.13$. (Down) velocity field at $t = 100$ projected on plane $x = 0.18$. Non-uniform mesh of $5 \times 20^2 \times 29 P_1-P_2$ tetrahedra.
{\medskip\noindent\bf References:}
[1] Chorin, A J, Numerical solution of the Navier-Stokes equations, {\it Math. Comp.}, {\bf 22}, (1968), 745-762 and {\bf 23}, (1969), 341-353.
[2] Deville M, L\^e T H and Morchoisne Y, Eds., {\it Numerical Simulation of 3-D Incompressible Unsteady Viscous Lminar Flows, Notes on Numerical Fluid Mechanics}, {\bf 36}, Vieweg, Wiesbaden, 1992.
[3] Goda, K, A multistep technique for the with implicit difference schemes for calculating two- or three-dimensional cavity flows, {\it J. Comput. Phys.}, {\bf 30}, (1979), 76-95.
[4] Guermond, J-L, Sur l'approximation des \'equations de Navier-Stokes instationnaires par une m\'ethode de projection. C. R. {\it Acad. Sc. Paris}, S\'erie I, {\bf 319}, (1994), 887-892.
[5] Guermond, J-L, Un r\'esultat de convergence h l'ordre deux en temps pour l'approximation des \'equations de Navier-Stokes par une technique de projection, to appear in {\it Mod\'el. Math. Anal. Num\'er}. (M$^2$ AN), (1998).
[6] Guermond, J-L and Quartapelle, L, Calculation of incompressible viscous flows by an unconditionally stable projection FEM, {\it J. Comput. Phys.}, {\bf 132}, (1997), 12-33.
[7] Rannacher, R, On Chorin's projection method for the incompressible Navier-Stokes equations, {\it Lectures Notes in Mathematics}, {\bf 1530}, Springer, Berlin, (1992). 167-183.
[8] Temam, H, Une m\'ethode d'approximation de la solution des \'equations de Navier-Stokes, {\it Bull. Soc. Math. France}, {\bf 98}, (1968), 115-152.