Primitive Variable Solvers for Conservative General Relativistic Magnetohydrodynamics

By Scott Noble, Charles F. Gammie, J. McKinney, L. Del Zanna. 2005

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Conservative numerical schemes for general relativistic magnetohydrodynamics (GRMHD) require a method for transforming between "conserved" variables such as momentum and energy density and "primitive" variables such as rest-mass density, internal energy, and components of the four-velocity. The forward transformation (primitive to conserved) has a closed-form solution, but the inverse transformation (conserved to primitive) requires the solution of a set of five nonlinear equations. Here we discuss the mathematical properties of the inverse transformation and present six numerical methods for performing the inversion. The first method solves the full set of five nonlinear equations directly using a Newton-Raphson scheme and a guess from the previous time step. The other methods reduce the five nonlinear equations to either one or two nonlinear equations that are solved numerically. Comparisons between the methods are made using a survey over phase space, a two-dimensional explosion problem, and a general relativistic MHD accretion disk simulation. The run time of the methods is also examined. Code implementing the schemes is available with the electronic edition of the article.

Subfields: Astronomy and Astrophysics, Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/a7386ef3fbf03e01241bf36aac9edc2daafdd60d

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Citations in the corpus

Simulations of magnetized discs around black holes: Effects of black hole spin, disc thickness and magnetic field geometry (2010, methodology): “…code is that we use horizon-penetrating KS coordinates for the Kerr metric (Gammie, McKinney & Tóth 2003; McKinney & Gammie 2004; McKinney 2006a; Noble et al. 2006; Mignone & McKinney 2007; Tchekhovskoy, McKinney & Narayan 2007), which avoids any issues with the coordinate singularity in BL…”
PLUTO: A Numerical Code for Computational Astrophysics (2007, methodology): “Thecomponents of U are related to the primitive variables V ( (cid:6); v ; p ; B ) through The inverse map is highly nonlinear and different methods of solution have been proposed when a constant (cid:2) law is adopted (for a recent review, see Noble et al. 2006).”
IPOLE - semi-analytic scheme for relativistic polarized radiative transport (2017, methodology): “The underlying plasma accretion flow model is a 3D GRMHD Fishbone-Moncrief torus simulation carried out with harm3d code (Gammie et al. 2003, Noble et al. 2006).”
A public code for general relativistic, polarised radiative transfer around spinning black holes (2016, methodology): “The simulation used the public ver- MNRAS 000, 1–22 (2015) sion of the HARM code (Gammie, McKinney & Tóth 2003; Noble et al. 2006) with a black hole spin of a = 0.9375.”
General relativistic magnetohydrodynamic simulations of magnetically choked accretion flows around black holes (2012, background): “…(with no contact steepener, but with shock flattener based upon rest-mass flux density and total pressure) for primitive quantities ({ρ0, ug, ũi, Bi}), a staggered magnetic field representation, and any regular grid warping (Gammie et al. 2003; Noble et al. 2006; Tchekhovskoy et al. 2007).”
A Reconnection Switch to Trigger gamma-Ray Burst Jet Dissipation (2010, methodology): “…steady-state jet solution is presented as based upon approximate analytical solutions that have been confirmed numerically (Tchekhovskoy et al. 2008, 2009, 2010b) using the HARM code (Gammie, McKinney & Tóth 2003; Noble et al. 2006; Mignone & McKinney 2007; Tchekhovskoy, McKinney & Narayan 2007).”
Three-dimensional relativistic MHD simulations of active galactic nuclei jets: magnetic kink instability and Fanaroff–Riley dichotomy (2015, methodology): “We carry out the simulations with the harm code (Gammie, McKinney, & Tóth 2003; Noble et al. 2006; Tchekhovskoy et al. 2007; McKinney & Blandford 2009; Tchekhovskoy et al. 2011 and use modified spherical polar coordinates (r, θ, ϕ) described below that span the range (rin, rout) × (0, π) × (0, 2π).”
Three-dimensional general relativistic radiation magnetohydrodynamical simulation of super-Eddington accretion, using a new code HARMRAD with M1 closure (2013, methodology): “…uses a single non-linear equation as a function of one MHD variable plus the MHD conserved quantities in order to obtain a solution to gas primitives Pn+1q (U n+1 q ) for the explicit case (Noble et al. 2006; Mignone & McKinney 2007; Sa̧dowski et al. 2013c) or steps within the implicit solver.”
BLACK HOLE SPIN AND THE RADIO LOUD/QUIET DICHOTOMY OF ACTIVE GALACTIC NUCLEI (2009, methodology): “We performed the simulations using the GR MHD code HARM (Gammie et al. 2003; McKinney & Gammie 2004; McKinney 2006a, 2006b; Noble et al. 2006) using Kerr– Schild coordinates in the Kerr metric; the code includes a number of recent improvements (Mignone & McKinney 2007; Tchekhovskoy et al. 2007,…”
Semi-implicit scheme for treating radiation under M1 closure in general relativistic conservative fluid dynamics codes (2012, methodology): “The conversion of hydrodynamical quantities is performed in the usual manner (e.g. Noble et al. 2006; Del Zanna et al. 2007).”
Formation of precessing jets by tilted black hole discs in 3D general relativistic MHD simulations (2017, methodology): “We developed H-AMR based on a 2D serial open-source code HARM2D (Gammie, McKinney & Tóth 2003; Noble et al. 2006).”
The Athena++ Adaptive Mesh Refinement Framework: Design and Magnetohydrodynamic Solvers (2020, methodology): “Noble et al. (2006) catalog six root-finding procedures used for variable inversion— four one-dimensional, one two-dimensional, and one fivedimensional, with the lower-dimensional versions solving for an enthalpy-like variable and/or a velocity-like variable, and with some of them only working for…”
Long-term GRMHD simulations of neutron star merger accretion discs: implications for electromagnetic counterparts (2018, methodology): “To evolve the torus in GRMHD, we use HARMPI2, an enhanced version of the serial open-source code HARM (Gammie et al. 2003; Noble et al. 2006).”
Equation of state in relativistic magnetohydrodynamics: variable versus constant adiabatic index (2007, methodology): “Noble et al. (2006) review several methods of inversion for the constant Γ-law, for which ρǫ = p/(Γ−1).”
Electron Thermodynamics in GRMHD Simulations of Low-Luminosity Black Hole Accretion (2015, methodology): “We implemented the model described in§2 and§3 into the conservative GRMHD code, HARM2D (High-Accuracy Relativistic Magnetohydrodynamics;Gammie, McKinney & Tóth 2003; Noble et al. 2006).”
ECHO: a Eulerian conservative high-order scheme for general relativistic magnetohydrodynamics and magnetodynamics (2007, methodology): “Here we basically follow the strategy outlined in Paper II; see also Noble et al. (2006) for further discussion and comparison of different techniques.”
THE SUBMILLIMETER BUMP IN Sgr A* FROM RELATIVISTIC MHD SIMULATIONS (2010, methodology): “For comparison, we have also run a set of axisymmetric simulations using the publicly available HARM code (Gammie et al. 2003; Noble et al. 2006) at black hole spins a/M = 0.00, 0.50, 0.75, 0.90, 0.92, and 0.94 (G = c = 1 is used throughout this paper).”