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REVIEW 4 major objections 4 minor 276 references

Smoothed particle magnetohydrodynamics for simulations of galaxy and cosmic structure formation

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This paper introduces a smoothed particle magnetohydrodynamics (SPMHD) formulation in the SWIFT code, and reports the first coupling of the EAGLE galaxy formation model to an MHD solver, with tests spanning a proto-stellar jet, a cluster…

desk verdict Solid SPMHD methods paper with honest validation, but the production-stability claim rests on an under-analyzed RMS beta estimator that can fail in exactly the low-beta clumps it is meant to protect. read the letter →

arxiv 2608.06978 v1 pith:X4LCYJ5R submitted 2026-08-07 astro-ph.GA astro-ph.COastro-ph.IM

Orestis A. Karapiperis , Matthieu Schaller , Nikyta Shchutskyi , Federico A. Stasyszyn , Maarten Elion This is my paper · ORCID
classification astro-ph.GAastro-ph.COastro-ph.IM MSC 65M7576W0585-08
keywords magnetohydrodynamicssmoothedparticlehydrodynamicsgalaxyformationmethods:numericalmagneticfieldsSWIFTEAGLEmodelcosmologicalsimulations
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out to make magnetic fields a standard component of production galaxy-formation simulations. It introduces a smoothed particle magnetohydrodynamics (SPMHD) formulation in the SWIFT code, built on the SPHENIX density-energy hydrodynamics scheme, with adaptive stabilisation: a tensile-instability correction whose strength is set by a new RMS-based local plasma $\beta$ estimate, a constrained hyperbolic/parabolic divergence cleaner, and artificial resistivity tuned to limit excess diffusion in cosmological flows. The authors argue the scheme is stable, accurate, and cheap enough to couple to effective sub-grid galaxy-formation recipes, and they demonstrate it on a jet-launching proto-stellar core, a massive galaxy cluster dynamo, and a Milky Way-like disk galaxy. The disk run is the first reported coupling of the EAGLE galaxy formation model to a magnetohydrodynamics solver. If the central claim is right, magnetic physics can be included in large cosmological runs at production cost.

What carries the argument

The load-bearing object is the SPMHD evolution system in SWIFT, expressed in a density-energy conservative form and paired with the SPHENIX discontinuity-capturing terms. Three mechanisms carry the stability argument: (i) a tensile-instability correction that subtracts a fraction of the monopole force, with the fraction set by the new estimator $\beta_{\rm loc} = \sqrt{\sum_j \beta_j^2 / \sum_j 1}$ (an unweighted RMS over neighbours, equation 47); (ii) a constrained hyperbolic/parabolic divergence-cleaning scalar whose cleaning speed is $c_h = v_{\rm sig}/2$ and whose damping rate has $\sigma_p = 1$ for critical damping; and (iii) artificial resistivity with signal velocity equal to the Alfvén speed and switch $\alpha_{\rm AR}=\min(\alpha_{\rm max}, h\|\nabla\mathbf{B}\|/\|\mathbf{B}\|)$, which is invariant under field rescaling. The design goal is that each term acts only where needed, limiting spurious dissipation while keeping particles stable in the disordered arrangements produced by sub-grid feedback.

What would settle it

Reproduce the EAGLE disk-galaxy application twice, once with the RMS plasma beta estimate (equation 47) and once with the naive per-particle beta (equation 46) driving the tensile-instability correction; the paper's claim predicts the naive version produces violent particle accelerations or ejections in feedback-disordered regions while the RMS version stays stable.

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Extended reading notes

Core claim

The paper's central claim is that a conservative density-energy SPMHD scheme, augmented with three adaptive regularisation ingredients, is stable, accurate, and computationally efficient enough for production galaxy-formation simulations. The ingredients are a tensile-instability correction modulated by an unweighted root-mean-square of neighbouring plasma betas rather than the particle's own $\beta$; a constrained mixed hyperbolic/parabolic divergence-cleaning scheme with cleaning speed set to half the pairwise signal velocity and critical damping at $\sigma_p=1$; and artificial resistivity with the Alfvén speed as signal velocity and a Tricco-Price shock indicator that is invariant under $\mathbf{B}\to\lambda\mathbf{B}$. With all hyperparameters fixed across the test suite, the method reproduces standard laboratory MHD experiments, launches a jet from a forming proto-stellar core, amplifies a magnetic field in a massive galaxy cluster, and evolves magnetic fields in a Milky Way-like disk galaxy, constituting the first reported EAGLE-MHD simulation.

Load-bearing premise

The load-bearing premise is that the new RMS-based local plasma beta estimate reliably tells the tensile-instability correction where to act, because if it fires in disordered, feedback-driven particle configurations the correction becomes a spurious repulsive force and the scheme becomes unstable.

Editorial extensions

If this is right

  • A galaxy-formation simulation with full sub-grid physics can now carry a magnetic field, so EAGLE-style runs gain a magnetised baseline for disk, cluster, and proto-stellar studies.
  • Because hyperparameters stay fixed across the entire test suite, the scheme's published configuration is a directly usable production setup rather than a per-problem tuned one.
  • The same solver handles laboratory shocks, jets, cluster dynamos, and disks, so a single code path can cover cosmological and object-scale MHD without switching methods.
  • Second-order convergence on smooth Alfvén waves and beyond-second-order convergence on Ohmic diffusion tests mean resolution studies with this scheme improve accuracy predictably.
  • The reported cluster and disk runs provide concrete magnetic field strengths and topologies that future cosmological MHD simulations can be compared against.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • We infer that the RMS plasma beta estimator, being less sensitive to disordered particle neighbourhoods, could also suppress spurious tensile corrections in other meshless MHD solvers that struggle with sub-grid feedback, though the paper only tests it in SWIFT.
  • A testable extension not explored in the paper is to compare long-term magnetic energy growth in cluster runs with and without the divergence cleaner's energy-conserving terms; the paper's energy accounting implies that growth should come from physical dynamo action rather than cleaning artefacts.
  • Because the divergence-cleaning speed is chosen as half the local signal speed rather than the fast magnetosonic speed, the scheme may be cheaper in cosmological boxes; a direct benchmark against the more common choice would quantify that gain.
  • The first EAGLE-MHD disk result suggests that magnetic fields can now be included as a standard module in future large-volume cosmological campaigns, with the main open question being the trade-off in computational cost as resolution and box size grow.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The manuscript presents a new smoothed particle magnetohrodynamics (SPMHD) formulation implemented in the open-source SWIFT code. The scheme is built on the SPHENIX density-energy SPH solver and adds a direct-induction ideal MHD discretisation, a tensile-instability correction with a novel RMS-based local plasma-beta estimator (Eq. 47), a constrained hyperbolic/parabolic divergence-cleaning scheme, artificial resistivity with an Alfvén-speed signal, Ohmic diffusion, and a cosmological comoving formulation. The authors validate the method on a comprehensive suite of standard tests: Alfvén wave convergence, MHD shock tubes, Orszag-Tang vortex, magnetic rotor, blast wave, Kelvin-Helmholtz and cloud-wind interaction, plus non-ideal diffusion tests. They announce three astrophysical applications (proto-stellar jet launching, galaxy-cluster dynamo, and a Milky Way-like disk with the EAGLE model) as the final part of the test suite. The full text provided to the referee, however, breaks off in Section 3.2.3 and does not contain Sections 3.3–3.5; the announced production-scale applications are therefore not accessible for assessment.

Significance. If the claims hold, this is a valuable contribution: the scheme is implemented in a modern, massively parallel, open-source code; the hyperparameters are kept fixed across the test suite; the validation set is broad and mostly quantitative; and the Alfvén-wave and non-ideal diffusion tests show second-order or better convergence. A successful coupling of the EAGLE galaxy-formation model to an MHD solver would be a first and would open the door to production magnetised galaxy-formation simulations. The central production-stability claim, however, rests on a new regularisation ingredient whose failure mode is not isolated by the presented tests, and the evidence that would demonstrate the claim in the target regime is in the missing Sections 3.3–3.5. The manuscript is therefore scientifically promising but, in its current provided form, does not yet support the headline application-level claims.

major comments (4)
  1. [Sections 3.3–3.5] The abstract and introduction announce three production-scale applications, culminating in 'the first reported coupling of the EAGLE galaxy formation model to a magnetohydrodynamics solver', but the provided manuscript text ends in the middle of Section 3.2.3 and never presents Sections 3.3, 3.4, or 3.5. The central claim of production viability, and specifically the stable operation of Eq. (47) in full-physics runs with sub-grid feedback, is therefore currently unsupported by any presented evidence. This is not a stylistic issue: the announced applications are the load-bearing demonstration that the method works in the regime for which it was designed. The revision must include these sections, with quantitative diagnostics (e.g., stability over time, divergence-error statistics, clumping checks) rather than only field maps.
  2. [Section 3.1.2, Eq. (44)–(45)] The Alfvén-wave test text states that the chosen parameters 'translate into a β < 1' and that the test assesses the tensile-instability correction in the strong-field regime. With the stated parameters, P = 0.1, B = 0.1, and μ0 = 1, the plasma beta is β = 2μ0P/B^2 = 20, not β < 1. The test therefore does not exercise the β < 2 branch of the switch λ(β_loc) in Eq. (44), and the claim that the scheme is 'robust to particle clumping in the strong field regime' is not demonstrated by this experiment. Either the parameters must be changed so that the low-β branch is actually probed, or the text must be corrected to state that the test covers only the β > 10 regime of λ.
  3. [Section 2.3.3, Eq. (47)] The RMS-based estimator β_loc (Eq. 47) is an unweighted average over all neighbours and is not the local plasma beta that appears in the stability criterion (43). In a magnetically dominated clump (β_i < 1) embedded in a high-β ambient medium, the RMS over the neighbourhood can be ≫ 1, setting λ = 0 in Eq. (44) and switching off the tensile-instability correction precisely where the linear-stability condition requires it to be active. The authors motivate Eq. (47) by 'extensive experimentation' and by improved behaviour in disordered, feedback-stirred particle arrangements, but no analysis or targeted test is provided for the regime where the RMS and the particle-local beta disagree. The existing validation tests do not isolate this case: the shock tubes and blast wave use ordered or nearly ordered lattices, and the Alfvén wave (as noted above) actually runs at β = 20. The manuscript needs a targeted disordered two-phase test — e.g., a low-β clump in a high-β ambient medium with a randomised or feedback-like particle distribution — or a quantitative study of the β_loc distribution in the application runs. Without this, the central production-stability claim rests on an empirical assertion.
  4. [Section 2.3.6, Eqs. (74)–(75)] The new time-step conditions introduced for B and ψ are not written as well-defined mathematical expressions. Eq. (74) divides a vector by an expression ending in a dot, and Eq. (75) contains a similar dangling '·' in the denominator; it is unclear what vector or scalar operation is intended. This matters because the authors state in Section 3 that these conditions are disabled by default in the presented tests, but the conditions are part of the method description and will be used by adopters. The equations need to be made explicit, including the definition of the norm or componentwise operation.
minor comments (4)
  1. [Section 3.1.8] In the description of the Kelvin-Helmholtz set-up, the text says the central region's physical attributes 'are denoted by the subscript O'; this should be the subscript C (and similarly in the following sentence).
  2. [Section 3.1.9, Fig. 17 caption] The caption lists the strong-field run as 'β = 250' in the fourth and fifth rows; from the body text and the simulation description, this should be β = 25.
  3. [Section 3.2.2] The text refers repeatedly to 'Karapiperis & Schaller (2025)' and 'Shchutskyi et al. (2025)' for validation of cosmological non-ideal MHD and dynamo benchmarks, but these are not presented in this manuscript; since the provided text also omits the application sections, the reader cannot verify the claimed cross-validation.
  4. [General] The manuscript is unusually long and some figure captions repeat the full setup parameters verbatim from the text; shortening the captions and moving the fully detailed setup data to a table would improve readability.

Circularity Check

1 steps flagged · score 2.0 of 10

Two openly calibrated hyperparameters appear in the same tests used as their own validation, but the core derivation and the bulk of the benchmark suite are externally anchored; no load-bearing circularity.

  1. fitted input called prediction [Section 3.1.4, 'MHD shock tubes'; Eq. (63)]
    "Keeping the calibration of SPHENIX's shock-capturing scheme unchanged and employing the defaults suggested by Borrow et al. (2022), we ran variations of the Brio & Wu (1988) shock tube to settle on an optimal value for αARmax, the global normalisation parameter entering our artificial resistivity implementation."

    αARmax is the free amplitude of the artificial resistivity term (Eq. 63), and the Brio-Wu shock tube is the exact test on which it was tuned. The same test is then presented in the same section as a validation ('Numerical predictions obtained with SWIFT ... agree well with the reference'). The agreement on that one experiment is therefore partly by construction: the free parameter was chosen to make that experiment agree. This is a calibration rather than an independent prediction, and it is limited in scope because all other tests in the suite use the same fixed value and are compared against independent references (ATHENA runs, analytic solutions, or ODE reference solutions).

full rationale

The core derivation is not circular: the SPMHD equations of motion (37)-(38), the non-ideal Ohmic terms (40)-(41), the tensile-instability correction (45), and the divergence-cleaning pair (51)-(52) follow from a Lagrangian/action principle and standard SPH discretizations, with no target outcome used as an input. The two openly declared calibration choices — σ_p in Eq. (54) tuned on the monopole-advection run, and αARmax in Eq. (63) tuned on the Brio-Wu shock tube — are the only places where an input is chosen to make a specific test agree; the paper is transparent about both, explicitly identifying the Brio-Wu test as the calibration standard. Consequently, the agreement shown on those two tests is partly by construction, but it is not a claimed physical prediction and it does not carry the central stability/accuracy claim, which rests on fixed-parameter comparisons to external ATHENA results (Orszag-Tang vortex, rotor), the analytic circularly polarized Alfvén wave, the diffusive Gaussian pulse solutions, the diffusive Alfvén-wave dispersion relations, and the C-shock ODE reference. The RMS β-loc estimator (47) is an empirically motivated ansatz ('after extensive experimentation'), not a derived or predicted quantity; whether it can suppress the tensile-instability correction in low-β clumps inside disordered two-phase media is a legitimate robustness/correctness question, not a circularity. Companion self-citations (Karapiperis & Schaller 2025; Shchutskyi et al. 2025) are auxiliary and not load-bearing: the in-paper benchmarks stand on independent evidence. Overall the derivation chain is self-contained, with only a minor, local calibration coupling in the validation narrative.

Assumptions & free parameters 2 free parameters · 7 assumptions · 0 invented entities

The method relies on standard SPMHD building blocks plus two calibrated hyperparameters (sigma_p, alpha_ARmax) and one new load-bearing empirical prescription (RMS beta_loc). No new physical entities are introduced.

free parameters (2)
  • sigma_p = 1
    Parabolic divergence-cleaning prefactor in equation (54); calibrated on the 3D magnetic monopole advection test (Section 3.1.1) to achieve critical damping.
  • alpha_ARmax = 0.1
    Global normalization of artificial resistivity in equation (63); calibrated on the Brio-Wu shock tube test (Section 3.1.4), consistent with Hopkins & Raives (2016).
assumptions (7)
  • domain assumption The ideal MHD equations (30)-(33) plus Ohmic diffusion (39) are the correct continuum limit.
    Standard physics of a fully ionized, weakly resistive fluid; invoked throughout Section 2.3.
  • standard math Discrete equations from an SPH action minimization (Lagrangian (36)) yield conservative and stable discretizations.
    Standard SPMHD derivation following Price & Monaghan (2004b); used in Section 2.3.1.
  • domain assumption The tensile instability correction of Børve et al. (2001) with the switch (44) from Price et al. (2018) is a valid way to control clumping.
    Section 2.3.3; relies on linear stability analysis of mean-field SPMHD.
  • standard math The constrained hyperbolic/parabolic divergence cleaning of Tricco & Price (2012) conserves energy and keeps divergence errors bounded.
    Section 2.3.4; proven in Tricco & Price (2012) and Tricco et al. (2016a).
  • ad hoc to paper The RMS-based plasma beta estimate (47) is a robust indicator of tensile instability onset in disordered particle fields.
    New empirical prescription in Section 2.3.3; load-bearing for stability in production runs with sub-grid feedback.
  • domain assumption The comoving variable transformation (80)-(82) preserves the structure of the MHD equations and gives consistent wave speeds.
    Section 2.3.7; follows Brandenburg et al. (1996) and requires the transformation to leave thermodynamic relations form-invariant.
  • domain assumption The EAGLE galaxy formation sub-grid model (Schaye et al. 2015) is an appropriate background for MHD coupling.
    Invoked in Section 3.5 (truncated) and in the abstract; the sub-grid model's validity is taken from the prior literature.

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Cite this review

Pith. "Pith review of Smoothed particle magnetohydrodynamics for simulations of galaxy and cosmic structure formation." pith.science (2026). https://pith.science/paper/X4LCYJ5R

@misc{pith2026260806978,
  author       = {Pith},
  title        = {Pith review of: Smoothed particle magnetohydrodynamics for simulations of galaxy and cosmic structure formation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X4LCYJ5R}},
  note         = {Machine review of arXiv:2608.06978}
}
read the original abstract

We introduce a novel formulation of cosmological smoothed particle magnetohydrodynamics (SPMHD), designed to model magnetic field physics in a vast array of nonlinear astrophysical systems, and which we have implemented in the highly-parallel, entirely modular, and open-source simulation code SWIFT. Our numerical scheme is designed to offer optimal performance at a minimal computational cost, keep a low memory footprint, and most notably couple robustly to effective sub-resolution recipes of galaxy formation. This is achieved through expressing our evolution equations in a density-energy conservative form, and augmenting them with discontinuity-capturing terms tailored to high dynamic range simulations, which are further modulated by adaptive switches that drastically improve coupling to sub-grid models and limit spurious dissipation. We moreover present novel suggestions for the two major regularisation techniques used in modern SPMHD, namely a tensile instability correction and mixed hyperbolic/parabolic divergence-cleaning scheme, to ensure code stability in highly dynamical scenarios. We evaluate the performance of our method on a series of problems of increasing complexity, culminating in three astrophysical applications which have historically proven challenging for mesh-less methods: we study jet launching from a forming proto-stellar core, dynamo amplification in a massive galaxy cluster and magnetic field evolution in a Milky Way-like disk galaxy; the latter constitutes the first reported coupling of the EAGLE galaxy formation model to a magnetohydrodynamics solver. Keeping model hyperparameters fixed across our test suite to provide a transparent picture of our method's capabilities in production, we demonstrate sound performance and convergence with resolution on standard `laboratory' numerical experiments, as well as competitive capabilities in realistic applications.

Figures

Figures reproduced from arXiv: 2608.06978 by the authors.

Figure 1
Figure 1. Results of the three-dimensional magnetic monopole advection tests, for a set of simulations run with a range of parabolic divergence cleaning multiplicative prefactors 𝜎𝑝 drawn from the interval [0.2, 1.4]. Left: time evolution of the box-averaged dimensionless divergence error. As 𝜎𝑝 is increased, we observe a gradual transition from oscillatory to purely decaying behaviour, with the damping eventually becoming sl… view at source ↗
Figure 2
Figure 2. Results of the Circularly polarised Alfvén wave test at t=5, which corresponds to five wave periods. Left: transverse magnetic field component 𝐵2 plotted against the coordinate along the wave propagation direction 𝑥1 for all particles in the simulation. We show results at three different resolution levels (solid, coloured lines), comparing them to the problem’s known, exact solution (dashed black line). Our scheme’s… view at source ↗
Figure 4
Figure 4. Time evolution of the box-averaged magnetic energy ⟨𝑩 2 ⟩/2 (mea￾sured in units of its initial value) for the magnetic field loop advection test, for Δ = 1 (red lines) and Δ = 2 (green lines), at low (dashed lines) and at our fiducial (solid lines) resolution. The effect of numerical diffusion (after settling of the initial conditions) is small, and the relative decrease in ⟨𝑩 2 ⟩/2 is of the order at most of just a… view at source ↗
Figures from the paper (29 more)
Figure 3
Figure 3. Figure 3: Results for the magnetic field loop advection test. We show projec￾tions in the 𝑥 − 𝑦 plane of the norm of the magnetic field vector 𝑩 measured in units of its initial value within the loop (left column), and of the norm of the magnetic current 𝑱 = ∇ × 𝑩 (right column)…
Figure 5
Figure 5. Figure 5: Results for the Brio & Wu shock tube test at t=0.1. The initial state left of the discontinuity is 𝑆𝐿 ≡ {𝜌𝐿 , 𝑃𝐿 , 𝒗𝐿 , 𝑩𝐿 } = {1, 1, (0, 0, 0), (0, 1, 0) }, that to its right is 𝑆𝑅 ≡ {𝜌𝑅, 𝑃𝑅, 𝒗𝑅, 𝑩𝑅 } = {0.125, 0.1, (0, 0, 0), (0, −1, 0) }; we resolve these with 2 × (…
Figure 6
Figure 6. Figure 6: Results for the Ryu & Jones 1A shock tube test at t=0.08. The initial state left of the discontinuity is 𝑆𝐿 ≡ {𝜌𝐿 , 𝑃𝐿 , 𝒗𝐿 , 𝑩𝐿 } = {1, 20, (10, 0, 0), (5/ √ 4𝜋, 5/ √ 4𝜋, 0) }, that to its right is 𝑆𝑅 ≡ {𝜌𝑅, 𝑃𝑅, 𝒗𝑅, 𝑩𝑅 } = {1, 1, (−10, 0, 0), (5/ √ 4𝜋, 5/ √ 4𝜋, 0) }. …
Figure 7
Figure 7. Figure 7: Results for the Ryu & Jones 2A shock tube test at t=0.08. The initial state left of the discontinuity is 𝑆𝐿 ≡ {𝜌𝐿 , 𝑃𝐿 , 𝒗𝐿 , 𝑩𝐿 } = {1.08, 0.95, (1.2, 0.01, 0.5), (2/ √ 4𝜋, 3.6/ √ 4𝜋, 2/ √ 4𝜋) }, that to its right is 𝑆𝑅 ≡ {𝜌𝑅, 𝑃𝑅, 𝒗𝑅, 𝑩𝑅 } = {1, 1, (0, 0, 0), (2/ √ 4𝜋…
Figure 8
Figure 8. Figure 8: Results for the Orszag-Tang vortex test at 𝑡 = 0.5 (1 st and 3 rd rows) and 𝑡 = 1 (2 nd and 4 th rows), run in 3D at a resolution of 𝑛𝑥 ∼ 128 (1 st column), 𝑛𝑥 ∼ 256 (2 nd column) and 𝑛𝑥 ∼ 512 (3 rd column) particles along either of the two major sides of the computati…
Figure 9
Figure 9. Figure 9: Four method diagnostics in our highest resolution Orszag-Tang vortex run (𝑛𝑥 ∼ 512 resolution elements along either of the two major sides of the computational domain). We show, from left to right, projections at 𝑡 = 0.5 of the artificial viscosity switch normalised to…
Figure 10
Figure 10. Figure 10: Comparison of SWIFT and ATHENA (taken from Stone et al. 2008) solutions for the Orszag-Tang vortex problem at 𝑡 = 0.5. Left: Horizontal slices of thermal pressure 𝑃 through 𝑦 = 0.3125 (𝑃1; top) and 𝑦 = 0.427 (𝑃2; bottom), retrieved through the 2D projections we show i…
Figure 11
Figure 11. Figure 11: Results for the fast magnetic rotor test at 𝑡 = 0.15, run at our fiducial resolution of 𝑛𝑥,𝑦 ∼ 128. We show, going from left to right, contour plots of the density, thermal pressure, Mach number and magnetic pressure in a two-dimensional slice taken at half the depth …
Figure 12
Figure 12. Figure 12: Comparison of SWIFT and ATHENA (taken from Stone et al. 2008) solutions for the fast magnetic rotor problem at 𝑡 = 0.15, similar to what is presented in [PITH_FULL_IMAGE:figures/full_fig_p028_12.png]
Figure 13
Figure 13. Figure 13: Results for the 3D strong magnetised blast wave test, run in a cubic simulation domain at our fiducial resolution of 2 × 2003 particles. A spherical pulse of high thermal pressure is introduced at 𝑡 = 0 in the middle the box, which is permeated by a constant, uniform …
Figure 14
Figure 14. Figure 14: Results for the 2D strong magnetised blast wave test, run in a thin 3D simulation box at our fiducial resolution of 2 × (192 × 192 × 24) particles. A cylindrical pulse of high thermal pressure is introduced at 𝑡 = 0 in the middle the box, which is permeated by a const…
Figure 15
Figure 15. Figure 15: Results for the magnetic Kelvin-Helmholtz instability test at 𝑡 = 𝜏𝐾𝐻 (left column) and 𝑡 = 2𝜏𝐾𝐻 (right column), run at our fiducial reso￾lution of 𝑛𝑥,𝑂 = 256 particles along the outer stream region’s longer side. We contrast results for a control hydrodynamic run ini…
Figure 17
Figure 17. Figure 17: Results for the magnetic cloud-wind interaction test at 𝑡 = 5𝜏𝑐𝑐 (left column) and 𝑡 = 10𝜏𝑐𝑐 (right column), run at our fiducial resolution of 𝑛 = 128 particles in the ambient medium along the simulation domain’s short side (of length 𝐿 = 1 in our code units). The ass…
Figure 18
Figure 18. Figure 18: Time evolution of selected global simulation metrics for the cloud￾wind interaction test. We differentiate between results from simulations run with a different initial plasma beta by line style (we plot 𝛽 = ∞ results as solid lines, 𝛽 = 250 results as dashed lines, a…
Figure 19
Figure 19. Figure 19: Results for the magnetic pulse diffusion test, for a pulse along the 𝑧 direction with a profile that is 1D (top row) and 2D (bottom row), resolved in either case by 643 particles in a cubic box of unit side length. The 1 st , 2 nd and 3 rd columns show results at 𝑡 = …
Figure 20
Figure 20. Figure 20: Convergence study for both the 1D and 2D magnetic pulse dif￾fusion tests. We show an L1 error on 𝐵𝑧 (triangle markers) computed over all particles in a given simulation at 𝑡 = 6𝑡0, for each of the three resolution levels considered for either variant of the problem. W…
Figure 21
Figure 21. Figure 21: Results for the travelling, circularly polarised, diffusive Alfvén wave test run with magnetic diffusivity set to 𝜂 = 0.1 and all our corrective measures enabled. Left: Time evolution of the system across ∼ 4 wave periods, corresponding to t=4 in our arbitrary system …
Figure 22
Figure 22. Figure 22: The same as [PITH_FULL_IMAGE:figures/full_fig_p039_22.png]
Figure 23
Figure 23. Figure 23: Results for the non-isothermal, non-radiative, oblique C-Shock test at 𝑡 = 1. The initial discontinuity is established by joining in the middle of the simulation domain the left state 𝑆𝐿 ≡ {𝜌𝐿 , 𝑃𝐿 , 𝒗𝐿 , 𝑩𝐿 } = {0.4, 0.4, (3, 0, 0), ( √ 2/2, √ 2/2, 0) } with the righ…
Figure 24
Figure 24. Figure 24: Gravitational collapse of a 1M⊙ molecular cloud. Instantiated as a uniform sphere of constant mass density in a homogeneous diffuse atmosphere 360 times less dense, the cloud is initially set in rigid body rotation about the 𝑧 axis with a period of 𝑇cloud = 470 Myr, a…
Figure 25
Figure 25. Figure 25: Similar to [PITH_FULL_IMAGE:figures/full_fig_p045_25.png]
Figure 26
Figure 26. Figure 26 [PITH_FULL_IMAGE:figures/full_fig_p046_26.png]
Figure 27
Figure 27. Figure 27: Different views of the MHD Santa Barbara cluster (SBC) at 𝑧 = 0. We show, from left to right, projections of the gas surface density Σgas, the strength of the magnetic field ∥𝑩∥, and the ratio of magnetic to thermal pressure 𝑃mag/𝑃therm = 1/𝛽. The maps are centred on …
Figure 28
Figure 28. Figure 28: Radial profiles for the MHD SBC at 𝑧 = 0; quantities are binned and subsequently averaged in 25 logarithmically spaced intervals of distance from the cluster centre. We show, from top to bottom, dark matter and gas density, gas temperature, magnetic field strength, an…
Figure 29
Figure 29. Figure 29: Evolution of particles tied to the Santa Barbara cluster (SBC) in the matter density 𝜌 (measured in units of the critical density of the universe 𝜌𝑐) to magnetic field strength ∥𝑩∥ phase space, at the three consecutive redshifts 𝑧 = 2, 1, 0 (from left to right). The c…
Figure 30
Figure 30. Figure 30: Kinetic (top) and magnetic (bottom) power spectra 𝑃𝒗 and 𝑃𝑩 for the Santa Barbara cluster at the four consecutive redshifts 𝑧 = 2, 1, 0.5, 0 (as indicated by line colour). Spectra are are computed by first depositing √︁ 𝜌/2𝒗 and √︁ 1/2𝜇0𝑩 on 3D ‘zero-padded’ Cartesian…
Figure 31
Figure 31. Figure 31: Evolution of a magnetised Milky Way-like isolated disk galaxy; example reproduced from Nobels et al. (2024), initialised with a uniform trace magnetic field of 10−3 𝜇G confined to the disk plane, and run with both gravity and MHD enabled, coupled to the EAGLE (Schaye …
Figure 32
Figure 32. Figure 32: Evolution of different energy components in the galactic disk. We show, as a function of time, the median of specific turbulent kinetic energies 𝜀turb. (solid black line), the median of specific thermal energies 𝜀therm. (dashed black line) and the median of specific m…
Figure 33
Figure 33. Figure 33: Divergence errors present in our magnetised Milky Way-like isolated disk galaxy simulation. The plotting configuration and panel layout is identical to [PITH_FULL_IMAGE:figures/full_fig_p056_33.png]

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Reviewed August 10, 2026 · model on record in the stance chip above.