Revisiting hyperbolicity of relativistic fluids

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, ideal magnetohydrodynamics, and resistive magnetohydrodynamics 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 magnetohydrodynamics 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 resistive magnetohydrodynamics, 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 multifluid and elastic models, are also expected to be amenable to our treatment.

Figure 1

Pointwise convergence order (color coding) for TOV star simulations with a DG scheme. Top: Only the interior of the star is simulated with a pure DG method and analytic outer boundary conditions. The stellar surface (dashed line) is not inside the numerical domain. Middle: Realistic setup of the entire star including its surface. It is surrounded by a low density atmosphere ${\rho }_{\mathrm{atm}}={10}^{-8}{\rho }_{\max }$. A pure DG method is used for the simulation. Bottom: Realistic setup of the entire star including its surface. The DG method is extended by a WENO-5 limiting procedure.