An Implicit Lagrangean Code for Spherically Symmetric General Relativistic Hydrodynamics with an Approximate Riemann Solver

An implicit Lagrangian hydrodynamics code for general relativistic spherical collapse is presented. This scheme is based on an approximate linearized Riemann solver (Roe type scheme). This code is aimed especially at the calculation of the late phase of collapse-driven supernovae and the nascent neutron star, where there is a remarkable contrast between the dynamical time scale of the proto-neutron star and the diffusion time scale of neutrinos, without such severe limitation of the Courant condition at the center of the neutron star. Several standard test calculations have been done. Two other adiabatic simulations have also been done in order to test the performance of the code in the context of the collapse-driven supernovae. It is found that the time step can be extended far beyond the Courant limitation at the center of the neutron star. The details of the scheme and the results of these test calculations are discussed.Comment: AASTeX v4.0, 24 pages, 13 figures on request from [email protected], submitted to Ap

Equation of State in Numerical Relativistic Hydrodynamics

Relativistic temperature of gas raises the issue of the equation of state (EoS) in relativistic hydrodynamics. We study the EoS for numerical relativistic hydrodynamics, and propose a new EoS that is simple and yet approximates very closely the EoS of the single-component perfect gas in relativistic regime. We also discuss the calculation of primitive variables from conservative ones for the EoS's considered in the paper, and present the eigenstructure of relativistic hydrodynamics for a general EoS, in a way that they can be used to build numerical codes. Tests with a code based on the Total Variation Diminishing (TVD) scheme are presented to highlight the differences induced by different EoS's.Comment: To appear in the ApJS September 2006, v166n1 issue. Pdf with full resolution figures can be downloaded from http://canopus.cnu.ac.kr/ryu/ryuetal.pd

RAM: A Relativistic Adaptive Mesh Refinement Hydrodynamics Code

We have developed a new computer code, RAM, to solve the conservative equations of special relativistic hydrodynamics (SRHD) using adaptive mesh refinement (AMR) on parallel computers. We have implemented a characteristic-wise, finite difference, weighted essentially non-oscillatory (WENO) scheme using the full characteristic decomposition of the SRHD equations to achieve fifth-order accuracy in space. For time integration we use the method of lines with a third-order total variation diminishing (TVD) Runge-Kutta scheme. We have also implemented fourth and fifth order Runge-Kutta time integration schemes for comparison. The implementation of AMR and parallelization is based on the FLASH code. RAM is modular and includes the capability to easily swap hydrodynamics solvers, reconstruction methods and physics modules. In addition to WENO we have implemented a finite volume module with the piecewise parabolic method (PPM) for reconstruction and the modified Marquina approximate Riemann solver to work with TVD Runge-Kutta time integration. We examine the difficulty of accurately simulating shear flows in numerical relativistic hydrodynamics codes. We show that under-resolved simulations of simple test problems with transverse velocity components produce incorrect results and demonstrate the ability of RAM to correctly solve these problems. RAM has been tested in one, two and three dimensions and in Cartesian, cylindrical and spherical coordinates. We have demonstrated fifth-order accuracy for WENO in one and two dimensions and performed detailed comparison with other schemes for which we show significantly lower convergence rates. Extensive testing is presented demonstrating the ability of RAM to address challenging open questions in relativistic astrophysics.Comment: ApJS in press, 21 pages including 18 figures (6 color figures

Primitive Variable Solvers for Conservative General Relativistic Magnetohydrodynamics

Conservative numerical schemes for general relativistic magnetohydrodynamics (GRMHD) require a method for transforming between ``conserved'' variables such as momentum and energy density and ``primitive'' variables such as rest-mass density, internal energy, and components of the four-velocity. The forward transformation (primitive to conserved) has a closed-form solution, but the inverse transformation (conserved to primitive) requires the solution of a set of five nonlinear equations. Here we discuss the mathematical properties of the inverse transformation and present six numerical methods for performing the inversion. The first method solves the full set of five nonlinear equations directly using a Newton-Raphson scheme and a guess from the previous timestep. The other methods reduce the five nonlinear equations to either one or two nonlinear equations that are solved numerically. Comparisons between the methods are made using a survey over phase space, a two-dimensional explosion problem, and a general relativistic MHD accretion disk simulation. The run-time of the methods is also examined. Code implementing the schemes is available for download on the web.Comment: Accepted to ApJ, 33 pages, 8 figures (color and greyscale), 1 machine-readable table (tab2.txt), code available at http://rainman.astro.uiuc.edu/codelib, a high-resolution and full-color PDF version is located at http://rainman.astro.uiuc.edu/codelib/codes/pvs_grmhd/ms.pd

Collimated Jet or Expanding Outflow: Possible Origins of GRBs and X-Ray Flashes

We investigate the dynamics of an injected outflow propagating in a progenitor in the context of the collapsar model for gamma-ray bursts (GRBs) through two dimensional axisymmetric relativistic hydrodynamic simulations. Initially, we locally inject an outflow near the center of a progenitor. We calculate 25 models, in total, by fixing its total input energy to be 10^{51} ergs s^{-1} and radius of the injected outflow to be $7\times 10^7$ cm while varying its bulk Lorentz factor, $\Gamma_{0} = 1.05\sim 5$, and its specific internal energy, $\epsilon_0/c^2 = 0.1\sim 30$. The injected outflow propagates in the progenitor and drives a large-scale outflow or jet. We find a smooth but dramatic transition from a collimated jet to an expanding outflow among calculated models. The maximum Lorentz factor is, on the other hand, sensitive to both of $\Gamma_0$ and $\epsilon_0$; roughly $\Gamma_{\rm max} \sim \Gamma_0 (1+\epsilon_0/c^2)$. Our finding will explain a smooth transition between the GRBs, X-ray rich GRBs (XRRs) and X-ray Flashes (XRFs) by the same model but with different $\epsilon_0$ values.Comment: Comments 51 pages, 21 figures. accepted for publication in ApJ high resolution version is available at http://www.mpa-garching.mpg.de/~mizuta/COLLAPSAR/collapsar.htm

Type II critical phenomena of neutron star collapse

We investigate spherically-symmetric, general relativistic systems of collapsing perfect fluid distributions. We consider neutron star models that are driven to collapse by the addition of an initially "in-going" velocity profile to the nominally static star solution. The neutron star models we use are Tolman-Oppenheimer-Volkoff solutions with an initially isentropic, gamma-law equation of state. The initial values of 1) the amplitude of the velocity profile, and 2) the central density of the star, span a parameter space, and we focus only on that region that gives rise to Type II critical behavior, wherein black holes of arbitrarily small mass can be formed. In contrast to previously published work, we find that--for a specific value of the adiabatic index (Gamma = 2)--the observed Type II critical solution has approximately the same scaling exponent as that calculated for an ultrarelativistic fluid of the same index. Further, we find that the critical solution computed using the ideal-gas equations of state asymptotes to the ultrarelativistic critical solution.Comment: 24 pages, 22 figures, RevTeX 4, submitted to Phys. Rev.

Relativistic Hydrodynamics around Black Holes and Horizon Adapted Coordinate Systems

Despite the fact that the Schwarzschild and Kerr solutions for the Einstein equations, when written in standard Schwarzschild and Boyer-Lindquist coordinates, present coordinate singularities, all numerical studies of accretion flows onto collapsed objects have been widely using them over the years. This approach introduces conceptual and practical complications in places where a smooth solution should be guaranteed, i.e., at the gravitational radius. In the present paper, we propose an alternative way of solving the general relativistic hydrodynamic equations in background (fixed) black hole spacetimes. We identify classes of coordinates in which the (possibly rotating) black hole metric is free of coordinate singularities at the horizon, independent of time, and admits a spacelike decomposition. In the spherically symmetric, non-rotating case, we re-derive exact solutions for dust and perfect fluid accretion in Eddington-Finkelstein coordinates, and compare with numerical hydrodynamic integrations. We perform representative axisymmetric computations. These demonstrations suggest that the use of those coordinate systems carries significant improvements over the standard approach, especially for higher dimensional studies.Comment: 10 pages, 4 postscript figures, accepted for publication in Phys. Rev.