On the stability of projection methods for the incompressible Navier-Stokes equations based on high-order discontinuous Galerkin discretizations

The present paper deals with the numerical solution of the incompressible Navier-Stokes equations using high-order discontinuous Galerkin (DG) methods for discretization in space. For DG methods applied to the dual splitting projection method, instabilities have recently been reported that occur for coarse spatial resolutions and small time step sizes. By means of numerical investigation we give evidence that these instabilities are related to the discontinuous Galerkin formulation of the velocity divergence term and the pressure gradient term that couple velocity and pressure. Integration by parts of these terms with a suitable definition of boundary conditions is required in order to obtain a stable and robust method. Since the intermediate velocity field does not fulfill the boundary conditions prescribed for the velocity, a consistent boundary condition is derived from the convective step of the dual splitting scheme to ensure high-order accuracy with respect to the temporal discretization. This new formulation is stable in the limit of small time steps for both equal-order and mixed-order polynomial approximations. Although the dual splitting scheme itself includes inf-sup stabilizing contributions, we demonstrate that spurious pressure oscillations appear for equal-order polynomials and small time steps highlighting the necessity to consider inf-sup stability explicitly.Comment: 31 page

Kinematic effect in gravitational lensing by clusters of galaxies

Gravitational lensing provides an efficient tool for the investigation of matter structures, independent of the dynamical or hydrostatic equilibrium properties of the deflecting system. However, it depends on the kinematic status. In fact, either a translational motion or a coherent rotation of the mass distribution can affect the lensing properties. Here, light deflection by galaxy clusters in motion is considered. Even if gravitational lensing mass measurements of galaxy clusters are regarded as very reliable estimates, the kinematic effect should be considered. A typical peculiar motion with respect to the Hubble flow brings about a systematic error < 0.3%, independent of the mass of the cluster. On the other hand, the effect of the spin increases with the total mass. For cluster masses ~ 10^{15}M_{sun}, the effect of the gravitomagnetic term is < 0.04% on strong lensing estimates and < 0.5% in the weak lensing analyses. The total kinematic effect on the mass estimate is then < 1%, which is negligible in current statistical studies. In the weak lensing regime, the rotation imprints a typical angular modulation in the tangential shear distortion. This would allow in principle a detection of the gravitomagnetic field and a direct measurement of the angular velocity of the cluster but the required background source densities are well beyond current tecnological capabilities.Comment: 6 pages; accepted for publication in MNRA

The Stellar Dynamics of Omega Centauri

The stellar dynamics of Omega Centauri are inferred from the radial velocities of 469 stars measured with CORAVEL (Mayor et al. 1997). Rather than fit the data to a family of models, we generate estimates of all dynamical functions nonparametrically, by direct operation on the data. The cluster is assumed to be oblate and edge-on but mass is not assumed to follow light. The mean motions are consistent with axisymmetry but the rotation is not cylindrical. The peak rotational velocity is 7.9 km/s at 11 pc from the center. The apparent rotation of Omega Centauri is attributable in part to its proper motion. We reconstruct the stellar velocity ellipsoid as a function of position, assuming isotropy in the meridional plane. We find no significant evidence for a difference between the velocity dispersions parallel and perpendicular to the meridional plane. The mass distribution inferred from the kinematics is slightly more extended than, though not strongly inconsistent with, the luminosity distribution. We also derive the two-integral distribution function f(E,Lz) implied by the velocity data.Comment: 25 Latex pages, 12 Postscript figures, uses aastex, epsf.sty. Submitted to The Astronomical Journal, December 199