A new formulation of general-relativistic hydrodynamic equations using primitive variables

TL;DR

This paper introduces a novel formulation of general-relativistic hydrodynamic equations using primitive variables, enabling more efficient and accurate simulations of compact objects in strong gravitational fields.

Contribution

The authors derive covariant hydrodynamic equations in primitive variables within the 3+1 framework and demonstrate their effectiveness through numerical tests on compact objects.

Findings

The new formulation is numerically stable and convergent.

It significantly reduces computational time in simulations.

The code accurately reproduces known oscillation modes.

Abstract

We present the derivation of hydrodynamical equations for a perfect fluid in General Relativity, within the 3+1 decomposition of spacetime framework, using only primitive variables. Primitive variables are opposed to conserved variables, as defined in the widely used Valencia formulation of the same hydrodynamical equations. The equations are derived in a covariant way, so that they can be used to describe any configuration of the perfect fluid. Once derived, the equations are tested numerically. We implement them in an evolution code for spherically symmetric self-gravitating compact objects. The code uses pseudospectral methods for both the metric and the hydrodynamics. First, convergence tests are performed, then the frequencies of radial modes of polytropes are recovered with and without the Cowling approximation, and finally the performance of our code in the black hole collapse and migration tests are described. The results of the tests and the comparison with a reference corecollapse and neutron star oscillations code suggests that not only our code can handle very strong gravitational fields, but also that this new formulation helps gaining a significant amount of computational time in hydrodynamical simulations of smooth flows in General Relativity.