A NUMERICAL STUDY OF THE AXISYMMETRICCOUETTE–TAYLOR PROBLEM USING A FASTHIGH-RESOLUTION SECOND-ORDER CENTRAL SCHEME∗RAZ KUPFERMAN†SIAM J. SCI. COMPUT.c°1998 Society for Industrial and Applied MathematicsVol. 20, No. 3, pp. 858–877Abstract. We present a numerical study of the axisymmetric Couette–Taylor problem usinga finite difference scheme. The scheme is based on a staggered version of a second-order central-differencing method combined with a discrete Hodge projection. The use of central-differencingoperators obviates the need to trace the characteristic flow associated with the hyperbolic terms.The result is a simple and efficient scheme which is readily adaptable to other geometries and to morecomplicated flows. The scheme exhibits competitive performance in terms of accuracy, resolution,and robustness. The numerical results agree accurately with linear stability theory and with previousnumerical studies.Key words. central difference schemes, incompressible flow, Couette–Taylor problemAMS subject classifications. 65M06, 76U05, 76E10PII. S10648275973180091. Introduction. Despite several decades of progress, the accurate computa-tion of flow problems is still a challenging task. Sophisticated schemes have beendesigned to cope with a variety of physical problems. Sophisticated methods are in-herently difficult to apply, especially if they require additional adaptation for eachspecific problem. This is an obstacle that often prevents the use of modern methodsin practical applications, e.g., in mechanical or chemical engineering. It is the purposeof this paper to show the applicability of a simple, easy-to-implement, computation-ally efficient, and readily generalizable scheme for flow problems. The realizationand performance of the scheme are demonstrated on the well-studied axisymmetricCouette–Taylor system.Many modern finite difference methods used in flow computations are based onthe Godunov paradigm, where the time evolution of a piecewise-polynomial approxi-mation of the flow field is sought. Typically, this piecewise-polynomial approximationis reconstructed from its cell averages. In this context, we distinguish between twomain classes of methods: upwind and central methods.Upwind schemes evaluate averages over the same computational cells that wereused to construct the initial piecewise-polynomial elements. The computation of thetime evolution of the flow field requires the evaluation of fluxes along the cell in-terfaces, i.e., along the discontinuous breakpoints. Consequently, the characteristicspeeds along such interfaces must be taken into account. Special attention is requiredat those interfaces in which there is a combination of forward- and backward-goingwaves, where it is necessary to decompose the “Riemann fan” and determine the sep-arate contribution of each component by tracing the “direction of the wind.” It is theneed to trace characteristic fans, using exact or approximate Riemann solvers, thatgreatly complicates the upwind algorithms. The first-order Godunov upwind scheme∗Received by the editors March 3, 1997; accepted for publication (in revised form) July 28, 1997;published electronically October 20, 1998. This work was supported by the U.S. Department ofEnergy under contract DE-AC03-76SF-00098.http://www.siam.org/journals/sisc/20-3/31800.html†Mathematics Department, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, 50A-2152,Berkeley, CA 94720 ([email protected]).858A NUMERICAL STUDY OF THE COUETTE–TAYLOR PROBLEM 859[9] is the forerunner for all the other Godunov-type schemes [16, 24, 11, 22, 7]. Forincompressible flow, the upwind-Godunov scheme was combined with Chorin’s pro-jection technique [5] by Bell, Colella, and Glaz [2], E and Shu [8], and others. For areview, see [10, 17, 6] and the references therein.Central schemes differ from upwind schemes in their way of calculating averages.In central schemes averages are evaluated over cells on a staggered grid so the break-points between the piecewise-polynomial elements are now inside the computationalcells. Averages are now integrated over the entire Riemann fan, while the correspond-ing fluxes are evaluated at the smooth centers of the piecewise-polynomial elements.This method obviates the need for Riemann solvers resulting in simpler and fasterschemes. The first-order Lax–Friedrichs (LxF) scheme [15] is the canonical exampleof such central schemes. Like Godunov’s upwind scheme, it is based on a piecewise-constant approximation. The LxF scheme, however, introduces excessive numericalviscosity resulting in relatively poor resolution.Modern high-resolution central schemes were introduced by Nessyahu and Tadmor(NT) [21] as a second-order sequel to the LxF scheme in one spatial dimension. Theoriginal NT scheme, which was based on a piecewise-linear approximation, yieldeda considerable improvement in terms of resolution; at the same time, it retainedthe relatively simple form of central schemes. The NT scheme was then extended tohigher orders [20] and to several spatial dimensions [12]. This and related work [25, 27]convincingly demonstrated that central schemes offer a much simpler alternative toupwind schemes while retaining a comparable resolution.The central schemes mentioned above were introduced primarily for hyperbolicconservation laws, such as those governing compressible flow. The conservation lawsfor incompressible flow are additionally constrained by the incompressibility condi-tion, which makes the dynamics nonlocal. The two-dimensional Euler equations intheir vorticity formulation were treated along these lines by Levy and Tadmor, both insecond- and third-order versions [18]. The resolution obtained by the latter is remark-able. However, there are two major shortcomings to using the vorticity formulation:boundary conditions are hard to formulate, and the method is not easily extended tothree spatial dimensions.These problems were resolved by Kupferman and Tadmor (KT) [14], where in-compressible flow was calculated in a velocity formulation based on the projectionmethod. The new scheme was tested on the classical doubly periodic shear layerand on longitudinal flow in a channel. The performance was compared to that of anupwind scheme. The two methods are comparable in accuracy and resolution. Thenew scheme was further found to be immune to the formation of spurious vorticalstructures [3].The simplicity, accuracy, and resolution of the KT scheme make it a