Revisiting Hyperbolicity of Relativistic Fluids

TL;DR

This paper reexamines the hyperbolicity and well-posedness of various models of relativistic fluids, revealing that common formulations are often only weakly hyperbolic, which impacts the stability of numerical simulations in gravitational wave research.

Contribution

It introduces a dual-frame formalism for analyzing hyperbolicity in relativistic fluid models, providing simplified expressions and identifying issues in standard formulations.

Findings

GRHD has a simplified hyperbolic form.

Standard flux-balance law formulations of GRMHD are only weakly hyperbolic.

Alternative formulations can achieve strong hyperbolicity.

Abstract

Motivated by the desire for highly accurate numerical computations of compact binary spacetimes in the era of gravitational wave astronomy, we reexamine hyperbolicity and well-posedness of the initial value problem for popular models of general relativistic fluids. Our analysis relies heavily on the dual-frame formalism, which allows us to work in the Lagrangian frame, where computation is relatively easy, before transforming to the desired Eulerian form. This general strategy allows for the construction of compact expressions for the characteristic variables in a highly economical manner. General relativistic hydrodynamics (GRHD), ideal magnetohydrodynamics (GRMHD) and resistive magnetohydrodynamics (RGRMHD) are considered in turn. In the first case we obtain a simplified form of earlier expressions. In the second we show that the flux-balance law formulation used in typical numerical applications is only weakly hyperbolic and thus does not have a well-posed initial value problem. Newtonian ideal MHD is found to suffer from the same problem when written in flux-balance law form. An alternative formulation, closely related to that of Anile and Pennisi, is instead shown to be strongly hyperbolic. In the final case we find that the standard forms of RGRMHD, relying upon a particular choice of `generalized Ohm's law', are only weakly hyperbolic. The latter problem may be rectified by adjusting the choice of Ohm's law, but we do not do so here. Along the way weak hyperbolicity of the field equations for dust and charged dust is also observed. More sophisticated systems, such as multi-fluid and elastic models are also expected to be amenable to our treatment.