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arxiv: 1906.09708 · v1 · pith:X6WSMFXHnew · submitted 2019-06-24 · 🌌 astro-ph.HE

Development and Application of Numerical Techniques for General-Relativistic Magnetohydrodynamics Simulations of Black Hole Accretion

Christopher J. White This is my paper

Pith reviewed 2026-05-25 17:34 UTC · model grok-4.3

classification 🌌 astro-ph.HE
keywords general-relativistic magnetohydrodynamicsblack hole accretionnumerical techniquesRiemann solversconstrained transportmagnetically arrested disks
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The pith

Advanced Riemann solvers and staggered-mesh constrained transport enable high-accuracy GRMHD simulations of black hole accretion.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper describes the implementation of advanced numerical techniques for general-relativistic magnetohydrodynamics simulations. The key improvements are the adoption of sophisticated Riemann solvers and staggered-mesh constrained transport. These changes, along with attention to performance and scalability, make it possible to model black hole accretion flows more accurately. The approach is illustrated by simulations of magnetically arrested disks.

Core claim

The central claim is that these numerical techniques permit investigation of black hole accretion flows with unprecedented accuracy, as shown through the exploration of magnetically arrested disks.

What carries the argument

The combination of advanced Riemann solvers and staggered-mesh constrained transport for general-relativistic magnetohydrodynamics.

If this is right

  • Black hole accretion flows can be simulated with higher fidelity in strongly magnetized regions.
  • Magnetically arrested disk configurations become accessible for detailed study.
  • The methods support investigations that require both accuracy and computational efficiency.
  • Parallel scalability allows running larger or longer simulations on modern computing systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • These simulation capabilities could help clarify the role of magnetic fields in powering relativistic jets from black holes.
  • Applying the methods to different initial conditions might reveal new stable accretion states.
  • Further development could incorporate additional physics such as radiative transfer while maintaining numerical stability.

Load-bearing premise

The new numerical techniques do not suffer from significant discretization errors or instabilities when applied to strongly curved spacetime and highly magnetized flows.

What would settle it

A direct comparison between the code results and an exact analytic solution for a black hole accretion problem that shows substantial discrepancies would falsify the claim of unprecedented accuracy.

Figures

Figures reproduced from arXiv: 1906.09708 by Christopher J. White.

Figure 1.1
Figure 1.1. Figure 1.1: Performance per core running a 3D GRMHD simulation on a cluster. [PITH_FULL_IMAGE:figures/full_fig_p016_1_1.png] view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: 1+1 spacetime diagram of a volume element Ω and its bounding surfaces Σ [PITH_FULL_IMAGE:figures/full_fig_p020_2_1.png] view at source ↗
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: 2+0 spacetime diagram of a single cell and its bounding surfaces and edges. The spacetime [PITH_FULL_IMAGE:figures/full_fig_p023_2_2.png] view at source ↗
Figure 2.3
Figure 2.3. Figure 2.3: Schematic 1+1 spacetime diagrams of different HLL Riemann solvers’ wavefans. The solid [PITH_FULL_IMAGE:figures/full_fig_p029_2_3.png] view at source ↗
Figure 2.4
Figure 2.4. Figure 2.4: 2+0 spacetime diagram showing a timeslice of four cells. The [PITH_FULL_IMAGE:figures/full_fig_p031_2_4.png] view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: Convergence tests for special-relativistic hydrodynamics. Both Riemann solvers converge at [PITH_FULL_IMAGE:figures/full_fig_p035_3_1.png] view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: Convergence tests for special-relativistic MHD. Both Riemann solvers converge at second order, [PITH_FULL_IMAGE:figures/full_fig_p036_3_2.png] view at source ↗
Figure 3.3
Figure 3.3. Figure 3.3: Convergence tests for general-relativistic hydrodynamics. Both Riemann solvers converge at [PITH_FULL_IMAGE:figures/full_fig_p037_3_3.png] view at source ↗
Figure 3.4
Figure 3.4. Figure 3.4: Convergence tests for general-relativistic MHD. Both Riemann solvers converge at second order, [PITH_FULL_IMAGE:figures/full_fig_p038_3_4.png] view at source ↗
Figure 3.5
Figure 3.5. Figure 3.5: Komissarov shock tubes for special-relativistic MHD. Results are computed with HLLD on a [PITH_FULL_IMAGE:figures/full_fig_p039_3_5.png] view at source ↗
Figure 3.6
Figure 3.6. Figure 3.6: Density ρ for the spherical blast wave at t = 15 in Minkowski coordinates (left), snake coordinates transformed back to Minkowski (center left), and snake coordinates (center right). The right panels show the fractional difference ∆ρ/ρ between the first and second columns, where ∆ρ = ρsnake − ρMinkowski and white regions have a fractional difference of less than 1%. The top panels use HLLE, while the bot… view at source ↗
Figure 3.7
Figure 3.7. Figure 3.7: Similar to Figure 3.6 with MHD added. Density [PITH_FULL_IMAGE:figures/full_fig_p042_3_7.png] view at source ↗
Figure 3.8
Figure 3.8. Figure 3.8: Magnetized Bondi flow in Schwarzschild coordinates with parameters as described in the text. [PITH_FULL_IMAGE:figures/full_fig_p043_3_8.png] view at source ↗
Figure 3.9
Figure 3.9. Figure 3.9: Errors in gas pressure for hydrodynamical (left) and MHD (right) Bondi flow on various [PITH_FULL_IMAGE:figures/full_fig_p044_3_9.png] view at source ↗
Figure 3.10
Figure 3.10. Figure 3.10: Errors in density, azimuthal magnetic field, and radial and azimuthal velocities as functions of [PITH_FULL_IMAGE:figures/full_fig_p045_3_10.png] view at source ↗
Figure 3.11
Figure 3.11. Figure 3.11: Errors in density for 2D Fishbone–Moncrief hydrodynamical tori on various [PITH_FULL_IMAGE:figures/full_fig_p046_3_11.png] view at source ↗
Figure 3.12
Figure 3.12. Figure 3.12: Density ρ of a 3D Fishbone–Moncrief MHD torus with effective resolution 288 × 256 × 192. Shown are the initial conditions in the rθ-plane (left), the final state at t = 12,000 in the same plane (center), and the final state in the rφ-plane (right). The electron temperature roughly matches what Shiokawa et al. find, both in magnitude and in trend with radius, though they do not have the same kink at arou… view at source ↗
Figure 3.13
Figure 3.13. Figure 3.13: Spherically averaged, time-averaged profiles of various quantities in the 3D torus. These are [PITH_FULL_IMAGE:figures/full_fig_p048_3_13.png] view at source ↗
Figure 3.14
Figure 3.14. Figure 3.14: Azimuthal correlation lengths as functions of radius in the 3D torus runs. These are for density [PITH_FULL_IMAGE:figures/full_fig_p049_3_14.png] view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: Grid used for MAD simulations, up to level 3 refinement. In this poloidal slice of the inner 11 [PITH_FULL_IMAGE:figures/full_fig_p053_4_1.png] view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: Initial density of MAD torus, shown at level 2 refinement and two zoom levels. [PITH_FULL_IMAGE:figures/full_fig_p054_4_2.png] view at source ↗
Figure 4.3
Figure 4.3. Figure 4.3: Initial gas pressure of MAD torus, shown at level 2 refinement and two zoom levels. [PITH_FULL_IMAGE:figures/full_fig_p054_4_3.png] view at source ↗
Figure 4.4
Figure 4.4. Figure 4.4: Initial normal observer Lorentz factor of MAD torus, shown at level 2 refinement and two zoom [PITH_FULL_IMAGE:figures/full_fig_p055_4_4.png] view at source ↗
Figure 4.5
Figure 4.5. Figure 4.5: Initial magnetic pressure of MAD torus, shown at level 2 refinement and two zoom levels. [PITH_FULL_IMAGE:figures/full_fig_p055_4_5.png] view at source ↗
Figure 4.6
Figure 4.6. Figure 4.6: Initial plasma β of MAD torus, shown at level 2 refinement and two zoom levels. Overlaid are streamlines for the magnetic field. the level 2 simulation, changing polar resolution, polar boundary conditions, and numerical floors, in order to investigate what effects these have. Furthermore we ran a very high resolution simulation (level 3) to a time of t = 4000 M in order to further study convergence. 4.4… view at source ↗
Figure 4
Figure 4. Figure 4: shows the density in the poloidal ( [PITH_FULL_IMAGE:figures/full_fig_p056_4.png] view at source ↗
Figure 4.7
Figure 4.7. Figure 4.7: Poloidal (top) and equatorial (bottom) slices of density in MAD torus at time [PITH_FULL_IMAGE:figures/full_fig_p057_4_7.png] view at source ↗
Figure 4.8
Figure 4.8. Figure 4.8: Poloidal (top) and equatorial (bottom) slices of gas pressure in MAD torus at time [PITH_FULL_IMAGE:figures/full_fig_p057_4_8.png] view at source ↗
Figure 4.9
Figure 4.9. Figure 4.9: Poloidal (top) and equatorial (bottom) slices of normal observer Lorentz factor in MAD torus [PITH_FULL_IMAGE:figures/full_fig_p058_4_9.png] view at source ↗
Figure 4.10
Figure 4.10. Figure 4.10: Poloidal (top) and equatorial (bottom) slices of magnetic pressure in MAD torus at time [PITH_FULL_IMAGE:figures/full_fig_p058_4_10.png] view at source ↗
Figure 4.11
Figure 4.11. Figure 4.11: Poloidal (top) and equatorial (bottom) slices of plasma [PITH_FULL_IMAGE:figures/full_fig_p059_4_11.png] view at source ↗
Figure 4.12
Figure 4.12. Figure 4.12: Poloidal (top) and equatorial (bottom) slices of plasma [PITH_FULL_IMAGE:figures/full_fig_p059_4_12.png] view at source ↗
Figure 4.13
Figure 4.13. Figure 4.13: Poloidal slices of density in MAD torus at time [PITH_FULL_IMAGE:figures/full_fig_p060_4_13.png] view at source ↗
Figure 4.14
Figure 4.14. Figure 4.14: Poloidal slices of plasma β in MAD torus at time t = 2000 M. Shown are the fiducial run (left) and the variation with double resolution near the pole (right). Overlaid are streamlines for the magnetic field lying on this slice through the data [PITH_FULL_IMAGE:figures/full_fig_p061_4_14.png] view at source ↗
Figure 4.15
Figure 4.15. Figure 4.15: Poloidal slices of plasma σ in MAD torus at time t = 2000 M. Shown are the fiducial run (left) and the variation with double resolution near the pole (right). Overlaid are streamlines for the magnetic field lying on this slice through the data. 55 [PITH_FULL_IMAGE:figures/full_fig_p061_4_15.png] view at source ↗
Figure 4.16
Figure 4.16. Figure 4.16: Poloidal slices of density in MAD torus at time [PITH_FULL_IMAGE:figures/full_fig_p062_4_16.png] view at source ↗
Figure 4.17
Figure 4.17. Figure 4.17: Poloidal slices of plasma β in MAD torus at time t = 10,000 M. Shown are the fiducial run (left), the variation with reflecting walls (center), and the variation with looser floors (right). Overlaid are streamlines for the magnetic field lying on this slice through the data. region, as is seen in the fiducial runs at lower resolution (Figures 4.11 and 4.12). The direction of the change indicates this li… view at source ↗
Figure 4.18
Figure 4.18. Figure 4.18: Poloidal slices of plasma σ in MAD torus at time t = 10,000 M. Shown are the fiducial run (left), the variation with reflecting walls (center), and the variation with looser floors (right). Overlaid are streamlines for the magnetic field lying on this slice through the data. In the equatorial plane the effects of magnetized, low-density fluid bubbling up are more apparent at higher resolution. Indeed an… view at source ↗
Figure 4.7
Figure 4.7. Figure 4.7: Similar patterns can be seen with β ( [PITH_FULL_IMAGE:figures/full_fig_p063_4_7.png] view at source ↗
Figure 4.19
Figure 4.19. Figure 4.19: Poloidal (top) and equatorial (bottom) slices of density in MAD torus at time [PITH_FULL_IMAGE:figures/full_fig_p064_4_19.png] view at source ↗
Figure 4.20
Figure 4.20. Figure 4.20: Poloidal (top) and equatorial (bottom) slices of plasma [PITH_FULL_IMAGE:figures/full_fig_p065_4_20.png] view at source ↗
Figure 4.21
Figure 4.21. Figure 4.21: Poloidal (top) and equatorial (bottom) slices of plasma [PITH_FULL_IMAGE:figures/full_fig_p066_4_21.png] view at source ↗
Figure 4.22
Figure 4.22. Figure 4.22: Zoomed in version of Figure 4.19, showing poloidal (top) and equatorial (bottom) slices of [PITH_FULL_IMAGE:figures/full_fig_p067_4_22.png] view at source ↗
Figure 4.23
Figure 4.23. Figure 4.23: Zoomed in version of Figure 4.20, showing poloidal (top) and equatorial (bottom) slices of [PITH_FULL_IMAGE:figures/full_fig_p068_4_23.png] view at source ↗
Figure 4.24
Figure 4.24. Figure 4.24: Zoomed in version of Figure 4.21, showing poloidal (top) and equatorial (bottom) slices of [PITH_FULL_IMAGE:figures/full_fig_p069_4_24.png] view at source ↗
Figure 4.25
Figure 4.25. Figure 4.25: Mass accretion, energy accretion, and horizon-threading flux in the MAD simulations as func [PITH_FULL_IMAGE:figures/full_fig_p071_4_25.png] view at source ↗
Figure 4.26
Figure 4.26. Figure 4.26: Mass accretion, energy accretion, and normalized horizon-threading flux in the MAD simulations [PITH_FULL_IMAGE:figures/full_fig_p072_4_26.png] view at source ↗
Figure 4.27
Figure 4.27. Figure 4.27: Efficiency parameter η in the MAD simulations as a function of time for four resolutions. All simulations are sampled at a rate of ∆t = 10 M. of the infalling material. The run of this efficiency in the various simulations is shown in [PITH_FULL_IMAGE:figures/full_fig_p073_4_27.png] view at source ↗
Figure 4.25
Figure 4.25. Figure 4.25: The sign of the slope of the curves indicates the field lines near the black hole are not closing [PITH_FULL_IMAGE:figures/full_fig_p074_4_25.png] view at source ↗
read the original abstract

We describe the implementation of sophisticated numerical techniques for general-relativistic magnetohydrodynamics simulations in the Athena++ code framework. Improvements over many existing codes include the use of advanced Riemann solvers and of staggered-mesh constrained transport. Combined with considerations for computational performance and parallel scalability, these allow us to investigate black hole accretion flows with unprecedented accuracy. The capability of the code is demonstrated by exploring magnetically arrested disks.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript describes the implementation of general-relativistic magnetohydrodynamics (GRMHD) in the Athena++ framework, incorporating advanced Riemann solvers and staggered-mesh constrained transport. These techniques, combined with attention to performance and scalability, are claimed to enable black hole accretion simulations with unprecedented accuracy, as demonstrated through explorations of magnetically arrested disks (MADs).

Significance. If the accuracy improvements are substantiated, the work would offer a valuable, scalable GRMHD tool for the astrophysics community studying black hole accretion. The focus on computational performance is a practical strength for large-scale applications.

major comments (2)
  1. [Abstract] Abstract: the assertion of 'unprecedented accuracy' is not supported by any quantitative comparisons, convergence tests, or error analysis.
  2. [Demonstration section] Demonstration of MAD simulations: no L1/L2 error norms, convergence rates, or direct comparisons against analytic solutions (e.g., magnetized Bondi flow) or other GRMHD codes are reported, leaving the central accuracy claim untested.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive feedback. We address each major comment below regarding the substantiation of accuracy claims.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the assertion of 'unprecedented accuracy' is not supported by any quantitative comparisons, convergence tests, or error analysis.

    Authors: We agree that the abstract's phrasing of 'unprecedented accuracy' is not supported by quantitative evidence presented in the manuscript. This wording was intended to highlight the benefits of the advanced Riemann solvers and staggered constrained transport relative to more dissipative methods commonly used, but we acknowledge the claim requires qualification. We will revise the abstract to remove 'unprecedented' and instead describe the methods as enabling 'high-accuracy' simulations of black hole accretion. revision: yes

  2. Referee: [Demonstration section] Demonstration of MAD simulations: no L1/L2 error norms, convergence rates, or direct comparisons against analytic solutions (e.g., magnetized Bondi flow) or other GRMHD codes are reported, leaving the central accuracy claim untested.

    Authors: We agree that the demonstration section does not include L1/L2 norms, convergence studies, or code-to-code comparisons. The section's purpose is to illustrate the code's application to physically interesting MAD flows rather than to serve as a validation study. Validation tests for the Athena++ GRMHD module appear in separate references. We will revise the manuscript to explicitly reference those prior validation results and add a brief discussion of the expected benefits of the chosen numerical methods for the reported simulations. revision: yes

Circularity Check

0 steps flagged

No circularity; code implementation and demonstration paper with no derivation chain

full rationale

This is a methods paper describing the implementation of advanced Riemann solvers and staggered-mesh constrained transport in Athena++ for GRMHD. No physical predictions, fitted parameters, or first-principles derivations are claimed. The demonstration via MAD simulations does not reduce any result to its own inputs by construction, nor does it rely on self-citation load-bearing uniqueness theorems or ansatzes. The central claim of capability is supported by the code description itself and is independent of any circular reduction. This matches the default expectation of no significant circularity for implementation work.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the central claim rests on the correctness of standard GRMHD equations and numerical methods from prior literature.

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