Robust recovery of primitive variables in relativistic ideal magnetohydrodynamics

Open Access

Robust recovery of primitive variables in relativistic ideal magnetohydrodynamics

Wolfgang Kastaun, Jay Vijay Kalinani, and Riccardo Ciolfi

Phys. Rev. D 103, 023018 – Published 20 January 2021

Abstract

Modern simulation codes for general relativistic ideal magnetohydrodynamics are all facing a long-standing technical problem given by the need to recover fundamental variables from those variables that are evolved in time. In the relativistic case, this requires the numerical solution of a system of nonlinear equations. Although several approaches are available, none has proven completely reliable. A recent study comparing different methods showed that all can fail, in particular for the important case of strong magnetization and moderate Lorentz factors. Here, we propose a new robust, efficient, and accurate solution scheme, along with a proof for the existence and uniqueness of a solution, and analytic bounds for the accuracy. Further, the scheme allows us to reliably detect evolution errors leading to unphysical states and automatically applies corrections for typical harmless cases. A reference implementation of the method is made publicly available as a software library. The aim of this library is to improve the reliability of binary neutron star merger simulations, in particular in the investigation of jet formation and magnetically driven winds.

Received 4 May 2020
Revised 14 August 2020
Accepted 2 October 2020

DOI:https://doi.org/10.1103/PhysRevD.103.023018

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Open access publication funded by the Max Planck Society.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Astrophysical fluid dynamics Magnetohydrodynamics

Numerical relativity

Fluid DynamicsGravitation, Cosmology & Astrophysics

Authors & Affiliations

Wolfgang Kastaun ^1,2, Jay Vijay Kalinani ^3,4, and Riccardo Ciolfi ^5,4

¹Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Callinstrasse 38, 30167 Hannover, Germany
²Leibniz Universität Hannover, 30167 Hannover, Germany
³Università di Padova, Dipartimento di Fisica e Astronomia, Via Francesco Marzolo 8, I-35131 Padova, Italy
⁴INFN, Sezione di Padova, Via Francesco Marzolo 8, I-35131 Padova, Italy
⁵INAF, Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, I-35122 Padova, Italy

Article Text

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References

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Issue

Vol. 103, Iss. 2 — 15 January 2021

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Images

Figure 1
Master function $f$ for different regimes, combining velocities $v = 0$ and $v = 0.99$ (labeled “vlarge”), magnetic field scale $b = 0$ and $b = 2$ (“blarge”), internal specific energy $ε_{th} = 0$ (“cold”) and 10 (“hot”), where $ε_{th}$ denotes the difference from the zero-temperature case. Velocities are oriented orthogonally to the magnetic field, which is the most difficult case. In the top panel, the independent variable $μ$ is scaled to the initial root bracket $μ_{+}$ , and the function is scaled to the maximum value over the interval shown. The lower panel shows the behavior near the root ${\tilde{μ}}_{0}$ .
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Figure 2
Schematic overview of the recovery algorithm information flow and list of required equations. Arrows denote dependencies, including variables further up in the chain. Double arrows denote computations that require information about the EOS validity bounds; the one double arrow labeled “EOS” refers to evaluation of the EOS. Intermediate results obtained during evaluation of the master function in the last root finding iteration are denoted “Last call.” Square brackets refer to the list of fixed parameters (independent of $μ$ ) during root solving. The density $ρ_{bnd}$ stands for any one of the upper and lower EOS validity bounds.
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Figure 3
Relative error of reconstructed pressure, as a function of specific thermal energy and velocity (the latter in terms of $z = W v$ ). The results were obtained for the case of the hybrid EOS (see text), at fixed mass density $6 \times 10^{12} g / {cm}^{3}$ , demanding an accuracy $Δ = 10^{- 8}$ . The upper panel shows results for zero magnetic field, and the lower panel for the magnetically dominated case $b = 10$ . The solid lines mark the regions where expected errors related to rounding start to exceed those related to root solving.
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Figure 4
Number of calls to the EOS required to reconstruct the primitives to accuracy $Δ = 10^{- 8}$ , as a function of specific thermal energy and velocity (in terms of $z = W v$ ). The results were obtained for the case of the hybrid EOS (see text) at a mass density $ρ = 6 \times 10^{12} g / {cm}^{3}$ . The upper panel shows results for the magnetic scale $b = 0$ , and the lower panel for the magnetically dominated case $b = 10$ .
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Figure 5
Number of calls to the EOS required to reconstruct the primitives to accuracy $Δ = 10^{- 8}$ , as a function of magnetic scale $b$ and velocity (in terms of $z = W v$ ). The results were obtained for the case of the hybrid EOS (see text) at a mass density $ρ = 6 \times 10^{12} g / {cm}^{3}$ . The upper panel shows results for cold matter $ε_{th} = 10^{- 4}$ , and the lower panel for very hot matter $ε_{th} = 10$ . For comparison, we also show the magnetization as contour lines of $\log_{10} (B^{2} / P)$ . The red line marks a magnetic field strength of $B = 10^{16} G$ .
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Figure 6
Amplification factors $A_{q}$ , $A_{r}$ , and $A_{b}$ for the relative error of $ε$ , as defined in Eq. (80), for the example of cold matter obeying the MS1 EOS. Top panel: amplification as a function of density, for fixed magnetic field and velocity. Bottom panel: amplification versus velocity, for fixed density and no magnetic field.
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