Recovery Schemes for Primitive Variables in General-relativistic Magnetohydrodynamics

Abstract

General-relativistic magnetohydrodynamic (GRMHD) simulations are an important tool to study a variety of astrophysical systems such as neutron star mergers, core-collapse supernovae, and accretion onto compact objects. A conservative GRMHD scheme numerically evolves a set of conservation equations for 'conserved' quantities and requires the computation of certain primitive variables at every time step. This recovery procedure constitutes a core part of any conservative GRMHD scheme and it is closely tied to the equation of state (EOS) of the fluid. In the quest to include nuclear physics, weak interactions, and neutrino physics, state-of-the-art GRMHD simulations employ finitetemperature, composition-dependent EOSs. While different schemes have individually been proposed, the recovery problem still remains a major source of error, failure, and inefficiency in GRMHD simulations with advanced microphysics. The strengths and weaknesses of the different schemes when compared to each other remain unclear. Here we present the first systematic comparison of various recovery schemes used in different dynamical spacetime GRMHD codes for both analytic and tabulated microphysical EOSs. We assess the schemes in terms of (i) speed, (ii) accuracy, and (iii) robustness. We find large variations among the different schemes and that there is not a single ideal scheme. While the computationally most efficient schemes are less robust, the most robust schemes are computationally less efficient. More robust schemes may require an order of magnitude more calls to the EOS, which are computationally expensive. We propose an optimal strategy of an efficient three-dimensional Newton-Raphson scheme and a slower but more robust one-dimensional scheme as a fall-back. * NASA Einstein Fellow arXiv:1712.07538v2 [astro-ph.HE]

Numerical relativity in spherical coordinates: A new dynamical spacetime and general relativistic MHD evolution framework for the Einstein Toolkit

“…In future versions of the code, problems related to the primitive recovery could be handled by more modern algorithms, such as evolving the entropy S and using it to recover the pressure [42], or using different primitive recovery schemes, see [153] for an overview. Another attractive approach could be the use of physical-constraintpreserving methods [151,154,155].…”

Spritz: a new fully general-relativistic magnetohydrodynamic code

“…In the literature many methods have been presented in order to perform this step [53,54]. In the Spritz code we implemented both the 2D method used in WhiskyMHD [2] and the one presented in [53] and used in GRHydro.…”

General-relativistic Resistive Magnetohydrodynamics with Robust Primitive-variable Recovery for Accretion Disk Simulations

“…where ξ ≔ ρhΓ 2 . Provided that E i and B i are known, such a system can usually be reduced to one single equation (in ξ, p, or other scalar variables; see e.g., Noble et al 2006or Siegel et al 2018 and solved with a one-dimensional (1D) Newton-Raphson (NR) iteration (where 1D refers to the single scalar equation that has to be solved and not to a spatial dimension). This is the standard approach in BHAC for the solution of the ideal-GRMHD equations (van der Holst et al 2008;Keppens et al 2012;Porth et al 2017).…”

“…To retrieve the primitive variables, it is necessary to solve one or more nonlinear equations. The solution method for the nonlinear equations is essential and is often a bottleneck for both accuracy and computational costs (Noble et al 2006;Siegel et al 2018). For stiff systems such as the set of GRRMHD equations, where the electric field is dynamically important, the primitive variables depend nonlinearly on the electric field and vice versa, resulting in an additional complication in the primitive-variable recovery compared to ideal-GRMHD.…”