zbMATH — the first resource for mathematics

A second-order projection method for the incompressible Navier-Stokes equations. (English) Zbl 0681.76030
We describe a second-order projection method for the time-dependent, incompressible Navier-Stokes equations. We first solve diffusion- convection equations to predict intermediate velocities which are then projected onto the space of divergence-free vector fields. By introducing more coupling between the diffusion-convection step and the projection step we obtain a temporal discretization that is second-order accurate. Our treatment of the diffusion-convection step uses a specialized higher order Godunov method for differencing the nonlinear convective terms that provides a robust treatment of these terms at high Reynolds number. The Godunov procedure is second-order accurate for smooth flow and remains stable for discontinuous initial data, even in the zero-viscosity limit. We approximate the projection directly using a Galerkin procedure that uses a local basis for discretely divergence-free vector fields. Numerical results are presented validating the convergence properties of the method. We also apply the method to doubly periodic shear-layers to assess the performance of the method on more difficult applications.

MSC:
 76D05 Navier-Stokes equations for incompressible viscous fluids 65N99 Numerical methods for partial differential equations, boundary value problems
Full Text:
References:
 [1] Ladyzhenskaya, O.A., Mathematical problems in the dynamics of a viscous incompressible flow, (1963), Gordon & Breach New York · Zbl 0121.42701 [2] Fujita, H.; Kato, T., Arch. rat. mech. anal., 16, 269, (1964) [3] Temam, R., Navier-Stokes equations, (1984), Elsevier Science Amsterdam · Zbl 0572.35083 [4] Harlow, F.H.; Welch, J.E., Phys. fluids, 8, 2182, (1965) [5] Gresho, P.M.; Sani, R.L., Int. J. numer. methods fluids, 7, 1111, (1987) [6] Krzywicki, A.; Ladyzhenskaya, O.A., Soviet phys. dokl., 11, 212, (1966) [7] Chorin, A.J., Math. comput., 22, 745, (1968) [8] Chorin, A.J., Math. comput., 23, 341, (1969) [9] Chorin, A.J., Stud. num. anal., 2, 64, (1968) [10] Temam, R., Arch. rat. mech. anal., 32, 135, 377, (1969) [11] Kim, J.; Moin, P., J. comput. phys., 59, 308, (1985) [12] Van Kan, J., SIAM J. sci. statist. comput., 7, 870, (1986) [13] Colella, P., A multidimensional second order Godunov scheme for conservation laws, (), (unpublished) [14] Van Leer, B., Multidimensional explicit difference schemes for hyperbolic conservation laws, (), 493 [15] Bell, J.B.; Dawson, C.N.; Shubin, G.R., J. comput. phys., 74, 1, (1988) [16] Stephens, A.B.; Bell, J.B.; Solomon, J.M.; Hackerman, L.B., J. comput. phys., 53, 152, (1984) [17] Russell, T.F.; Wheeler, M.F., Finite element and finite difference methods for continuous flows in porous media, () · Zbl 0572.76089 [18] Weiser, A.; Wheeler, M., SIAM J. num. anal., 25, 351, (1988) [19] Saltzman, J.S., Monotone difference schemes for the linear advection equation in two and three dimensions, (), (unpublished) [20] Solomon, J.M.; Szymczak, W.G., Finite difference solutions for the incompressible navierstokes equations using Galerkin techniques, () [21] Bell, J.B.; Glaz, H.M.; Solomon, J.M.; Szymczak, W.G., Application of a second-order projection method to the study of shear layers, ()
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.