Survey of numerical methods for compressible fluids

The finite difference methods of Godunov, Hyman, Lax-Wendroff (two-step), MacCormack, Rusanov, the upwind scheme, the hybrid scheme of Harten and Zwas, the antidiffusion method of Boris and Book, and the artificial compression method of Harten are compared with the random choice known as Glimm's method. The methods are used to integrate the one-dimensional equations of gas dynamics for an inviscid fluid. The results are compared and demonstrate that Glimm's method has several advantages. 16 figs., 4 tables

Pion Interferometry for a Granular Source of Quark-Gluon Plasma Droplets

We examine the two-pion interferometry for a granular source of quark-gluon plasma droplets. The evolution of the droplets is described by relativistic hydrodynamics with an equation of state suggested by lattice gauge results. Pions are assumed to be emitted thermally from the droplets at the freeze-out configuration characterized by a freeze-out temperature $T_f$. We find that the HBT radius $R_{out}$ decreases if the initial size of the droplets decreases. On the other hand, $R_{side}$ depends on the droplet spatial distribution and is relatively independent of the droplet size. It increases with an increase in the width of the spatial distribution and the collective-expansion velocity of the droplets. As a result, the value of $R_{out}$ can lie close to $R_{side}$ for a granular quark-gluon plasma source. The granular model of the emitting source may provide an explanation to the RHIC HBT puzzle and may lead to a new insight into the dynamics of the quark-gluon plasma phase transition.Comment: 5 pages, 4 figure

New Relativistic Effects in the Dynamics of Nonlinear Hydrodynamical Waves

In Newtonian and relativistic hydrodynamics the Riemann problem consists of calculating the evolution of a fluid which is initially characterized by two states having different values of uniform rest-mass density, pressure and velocity. When the fluid is allowed to relax, one of three possible wave-patterns is produced, corresponding to the propagation in opposite directions of two nonlinear hydrodynamical waves. New effects emerge in a special relativistic Riemann problem when velocities tangential to the initial discontinuity surface are present. We show that a smooth transition from one wave-pattern to another can be produced by varying the initial tangential velocities while otherwise maintaining the initial states unmodified. These special relativistic effects are produced by the coupling through the relativistic Lorentz factors and do not have a Newtonian counterpart.Comment: 4 pages, 5 figure

An Euler Solver Based on Locally Adaptive Discrete Velocities

A new discrete-velocity model is presented to solve the three-dimensional Euler equations. The velocities in the model are of an adaptive nature---both the origin of the discrete-velocity space and the magnitudes of the discrete-velocities are dependent on the local flow--- and are used in a finite volume context. The numerical implementation of the model follows the near-equilibrium flow method of Nadiga and Pullin [1] and results in a scheme which is second order in space (in the smooth regions and between first and second order at discontinuities) and second order in time. (The three-dimensional code is included.) For one choice of the scaling between the magnitude of the discrete-velocities and the local internal energy of the flow, the method reduces to a flux-splitting scheme based on characteristics. As a preliminary exercise, the result of the Sod shock-tube simulation is compared to the exact solution.Comment: 17 pages including 2 figures and CMFortran code listing. All in one postscript file (adv.ps) compressed and uuencoded (adv.uu). Name mail file `adv.uu'. Edit so that `#!/bin/csh -f' is the first line of adv.uu On a unix machine say `csh adv.uu'. On a non-unix machine: uudecode adv.uu; uncompress adv.tar.Z; tar -xvf adv.ta

Three-dimensional CFD simulations with large displacement of the geometries using a connectivity-change moving mesh approach

This paper deals with three-dimensional (3D) numerical simulations involving 3D moving geometries with large displacements on unstructured meshes. Such simulations are of great value to industry, but remain very time-consuming. A robust moving mesh algorithm coupling an elasticity-like mesh deformation solution and mesh optimizations was proposed in previous works, which removes the need for global remeshing when performing large displacements. The optimizations, and in particular generalized edge/face swapping, preserve the initial quality of the mesh throughout the simulation. We propose to integrate an Arbitrary Lagrangian Eulerian compressible flow solver into this process to demonstrate its capabilities in a full CFD computation context. This solver relies on a local enforcement of the discrete geometric conservation law to preserve the order of accuracy of the time integration. The displacement of the geometries is either imposed, or driven by fluid–structure interaction (FSI). In the latter case, the six degrees of freedom approach for rigid bodies is considered. Finally, several 3D imposed-motion and FSI examples are given to validate the proposed approach, both in academic and industrial configurations

First order hyperbolic formalism for Numerical Relativity

The causal structure of Einstein's evolution equations is considered. We show that in general they can be written as a first order system of balance laws for any choice of slicing or shift. We also show how certain terms in the evolution equations, that can lead to numerical inaccuracies, can be eliminated by using the Hamiltonian constraint. Furthermore, we show that the entire system is hyperbolic when the time coordinate is chosen in an invariant algebraic way, and for any fixed choice of the shift. This is achieved by using the momentum constraints in such as way that no additional space or time derivatives of the equations need to be computed. The slicings that allow hyperbolicity in this formulation belong to a large class, including harmonic, maximal, and many others that have been commonly used in numerical relativity. We provide details of some of the advanced numerical methods that this formulation of the equations allows, and we also discuss certain advantages that a hyperbolic formulation provides when treating boundary conditions.Comment: To appear in Phys. Rev.

Numerical Evolution of General Relativistic Voids

In this paper, we study the evolution of a relativistic, superhorizon-sized void embedded in a Friedmann-Robertson-Walker universe. We numerically solve the spherically symmetric general relativistic equations in comoving, synchronous coordinates. Initially, the fluid inside the void is taken to be homogeneous and nonexpanding. In a radiation- dominated universe, we find that radiation diffuses into the void at approximately the speed of light as a strong shock---the void collapses. We also find the surprising result that the cosmic collapse time (the $1^{\rm st}$-crossing time) is much smaller than previously thought, because it depends not only on the radius of the void, but also on the ratio of the temperature inside the void to that outside. If the ratio of the initial void radius to the outside Hubble radius is less than the ratio of the outside temperature to that inside, then the collapse occurs in less than the outside Hubble time. Thus, superhorizon-sized relativistic void may thermalize and homogenize relatively quickly. These new simulations revise the current picture of superhorizon-sized void evolution after first-order inflation.Comment: 37 pages plus 12 figures (upon request-- [email protected]) LaTeX, FNAL-PUB-93/005-

Radiative Backreaction on Global Strings

We consider radiative backreaction for global strings using the Kalb-Ramond formalism. In analogy to the point electron in classical electrodynamics, we show how local radiative corrections to the equations of motion allow one to remove the divergence in the self field and calculate a first order approximation to the radiation backreaction force. The effects of this backreaction force are studied numerically by resubstituting the equations of motion to suppress exponentially growing solutions. By direct comparison with numerical field theory simulations and analytic radiation calculations we establish that the `local backreaction approximation' provides a satisfactory quantitative description of radiative damping for a wide variety of string configurations. Finally, we discuss the relevance of this work to the evolution of a network of global strings and their possible cosmological consequences. These methods can also be applied to describe the effects of gravitational radiation backreaction on local strings, electromagnetic radiation backreaction on superconducting strings and other forms of string radiative backreaction.Comment: 38 Pages, Plain TEX, to appear Phys. Rev. D. Figures not included. Hard copy available by email to [email protected]

Characteristic Evolution and Matching

I review the development of numerical evolution codes for general relativity based upon the characteristic initial value problem. Progress in characteristic evolution is traced from the early stage of 1D feasibility studies to 2D axisymmetric codes that accurately simulate the oscillations and gravitational collapse of relativistic stars and to current 3D codes that provide pieces of a binary black hole spacetime. Cauchy codes have now been successful at simulating all aspects of the binary black hole problem inside an artificially constructed outer boundary. A prime application of characteristic evolution is to extend such simulations to null infinity where the waveform from the binary inspiral and merger can be unambiguously computed. This has now been accomplished by Cauchy-characteristic extraction, where data for the characteristic evolution is supplied by Cauchy data on an extraction worldtube inside the artificial outer boundary. The ultimate application of characteristic evolution is to eliminate the role of this outer boundary by constructing a global solution via Cauchy-characteristic matching. Progress in this direction is discussed.Comment: New version to appear in Living Reviews 2012. arXiv admin note: updated version of arXiv:gr-qc/050809

EGF increases expression and activity of PAs in preimplantation rat embryos and their implantation rate

BACKGROUND: Embryo implantation plays a major role in embryogenesis and the outcome of pregnancy. Plasminogen activators (PAs) have been implicated in mammalian fertilization, early stages of development and embryo implantation. As in-vitro developing embryos resulted in lower implantation rate than those developed in-vivo we assume that a reduced PAs activity may be involved. In the present work we studied the effect of EGF on PAs activity, quantity and embryo implantation. METHODS: Zygotes were flushed from rat oviducts on day one of pregnancy and grown in-vitro in R1ECM supplemented with EGF (10 ng/ml) and were grown up to the blastocyst stage. The control groups were grown in the same medium without EGF. The distribution and quantity of the PAs were examined using fluorescence immunohistochemistry followed by measurement of PAs activity using the chromogenic assay. Implantation rate was studied using the embryo donation model. RESULTS: PAs distribution in the embryos was the same in EGF treated and untreated embryos. Both PAs were localized in the blastocysts' trophectoderm, supporting the assumption that PAs play a role in the implantation process in rats. EGF increased the quantity of uPA at all stages studied but the 8-cell stage as compared with controls. The tissue type PA (tPA) content was unaffected except the 8-cell stage, which was increased. The activity of uPA increased gradually towards the blastocyst stage and more so due to the presence of EGF. The activity of tPA did not vary with the advancing developmental stages although it was also increased by EGF. The presence of EGF during the preimplantation development doubled the rate of implantation of the treated group as compared with controls