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

Abstract

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) and needs no artificial viscosity. 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 timescale of the proto-neutron star and the diffusion timescale of neutrinos, without such severe limitation of the Courant condition at the center of the neutron star. Several standard test calculations have been done, and their results show the following: (1) This code captures the shock wave accurately, although some erroneous jumps of specific internal energy are found at the contact discontinuity in the shock tube problems. (2) The scheme shows no instability even if we choose the Courant number larger than 1. (3) However, the Courant number should be kept below ~0.2 at the shock position, so that the shock can be resolved with a few meshes. (4) The scheme reproduces the well-known analytic solutions to the point blast explosion, the gravitational collapse of the uniform gas with γ = 4/3, and the general relativistic collapse of uniform dust. In addition, what is more important, calculations of hydrostatic configurations and the onset of radial instability, a preliminary neutrino transport in a proto-neutron star, and adiabatic collapses of a stellar core have also been done in order to show 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, which fact is crucially significant for the purpose of this project. The details of the scheme and the results of these test calculations are discussed.