Pith. sign in

REVIEW 1 major objections 6 minor 65 references

A structure-preserving Numerical Method for the Compressible Resistive-Hall-MHD System

T0 review · 1 major / 6 minor · reviewed 2026-08-02 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A splitting scheme for compressible resistive Hall-MHD preserves total energy, positivity, entropy dissipation, and the magnetic divergence involution constraint exactly.

desk verdict A genuine first for compressible resistive Hall-MHD structure preservation, with solid derivations, but practical robustness rests on unquantified artificial-resistivity tuning. read the letter →

arxiv 2607.14286 v1 pith:VGGDW3GB submitted 2026-07-15 math.NA cs.NAphysics.comp-phphysics.plasm-ph

Murtazo Nazarov , Rafael Rodriguez-Velasco , Ignacio Tomas This is my paper · ORCID
classification math.NAcs.NAphysics.comp-phphysics.plasm-ph MSC 65M6065M1276W05
keywords structure-preservingmethodsHall-MHDresistiveMHDfiniteelementmethodcurl-conformingelementsoperatorsplittingmagneticreconnectionartificialresistivity
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

This paper presents a finite-element method for the compressible resistive Hall-MHD equations designed to preserve four physical structures exactly: total energy, positive density and internal energy, entropy dissipation, and the magnetic-field divergence involution constraint. The key idea is an operator split: a standard explicit hyperbolic solver advances the compressible Euler part, while a curl-conforming finite element space advances the Lorentz-force/induction source system implicitly with a Crank-Nicolson step. The paper proves these preservation properties in Proposition 3.2, analyzes the coercivity of the Newton Jacobian, and introduces an artificial resistivity based on the electron velocity to keep Newton's iteration well posed. It validates the method with near-second-order convergence on a resistive whistler wave, reproduces the GEM magnetic reconnection challenge at resolutions up to 1024x1024 elements, and reports the first macroscopic simulation of the fully compressible resistive Hall-MHD Orszag-Tang vortex. If correct, this gives the community a structure-preserving baseline for a model of magnetic reconnection in which the Hall term is essential.

What carries the argument

The argument rests on an operator split: Operator #1 is the compressible Euler system advanced by an explicit SSP-RK method; Operator #2 is the magnetic source system (Lorentz force coupled to the induction equation with ideal, Hall, and resistive terms) advanced implicitly by Crank-Nicolson. The magnetic field is discretized in the curl-conforming BDM1 finite element space, so grad W_h0 subset of H_h holds, and the divergence involution is preserved by taking the curl of a gradient to be zero. The energy update adds the Joule heating term r |curl H|^2 through a lumped projection, converting resistive dissipation into internal energy exactly. Newton's method for the implicit step is stabiliz

What would settle it

Run the Orszag-Tang vortex with d_i = 0.5 on the directionally-biased structured mesh using the stated CFL (0.05) and resistivity constants, and at every time step until t = 1 check (1) whether the total energy sum_i m_i E_i + mu/2 ||H||^2 changes beyond round-off, and (2) whether Newton's iteration reaches a residual below 1e-8 within 25 iterations. If either condition fails, the claims of exact energy conservation and mesh-robust operation are falsified.

Watch

Extended reading notes

Core claim

The paper's central claim is Proposition 3.2: the splitting update hall_mhd_update satisfies total conservation of energy (sum_i m_i E_i^{n+1} + mu/2 ||H^{n+1}||^2 = sum_i m_i E_i^n + mu/2 ||H^n||^2), admissibility (rho > 0 and E - 1/2 |m|^2/rho > 0), the entropy-dissipation inequality (sum_i m_i eta(u_i^{n+1}) <= sum_i m_i eta(u_i^n)), and the involution constraint (H^{n+1}, grad omega) = (H^n, grad omega) for all omega in W_h0, provided the underlying Euler solver preserves the corresponding properties. Numerical tests show near-second-order convergence (L2 rates 1.92-2.00) on a resistive whistler wave, GEM reconnection at up to 1024x1024 elements, and the first reported macroscopic simula

Load-bearing premise

The practical stability of the scheme rests on empirically tuned artificial resistivity constants (c_low = 0.25, c_res = 1.0) and per-regime CFL reductions; the proven coercivity bound in Appendix B requires a stronger resistivity than the scheme actually uses, so in untested Hall-dominated regimes Newton's iteration could fail even though the structure-preservation theorems remain true.

Editorial extensions

If this is right

  • No divergence cleaning is needed: the involution constraint on the magnetic field is preserved by construction, on both structured and unstructured meshes.
  • Resolutions up to 1024x1024 elements for GEM reconnection and 724x724 for the Orszag-Tang vortex provide reference-quality data for a model that has mostly been simulated on 128x128 cells.
  • Because the induction equation receives no stabilization beyond the artificial resistivity, the scheme remains compatible with a vanishing-resistivity interpretation of MHD solutions.
  • The coercivity estimates give explicit sufficient conditions (small time step, or r_min > 1/2 mu c_e) under which the Newton Jacobian is guaranteed invertible, providing a principled guide for time-step selection.
  • Entropy dissipation is inherited from any Euler solver that satisfies a discrete entropy inequality, so the structure-preservation result transfers to a family of hyperbolic solvers.

Reading between the lines

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

  • A natural extension not pursued in the paper would be adaptive mesh refinement driven by the Jacobian coercivity estimate, which could replace the ad hoc CFL reductions used for d_i = 0.25 and 0.5 in the Orszag-Tang runs.
  • The proof of structure preservation is dimension-agnostic, so the framework should extend to fully three-dimensional Hall-MHD; the practical obstacle is the cost of Newton's method in 3D, not the discretization design.
  • The artificial resistivity is a nonlinear diffusion based on the electron velocity, so similar stabilization could apply in other plasma models where the electron speed far exceeds the ion speed.
  • If the Orszag-Tang results here are reproduced by independent codes, they could serve as a standard verification benchmark for the compressible resistive Hall-MHD equations.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 6 minor

Summary. The paper presents a second-order (in benchmarks) operator-splitting finite element scheme for the compressible resistive Hall-MHD system. The PDE is split into a compressible Euler operator and a magnetic source operator; the Euler part uses continuous P1 elements with SSP-RK, while the magnetic part uses curl-conforming BDM-like elements and Crank-Nicolson time stepping, with Newton iteration. The main theoretical results are Proposition 3.2: the complete split scheme conserves total energy, preserves admissible states, dissipates mathematical entropy, and preserves the divergence involution, under stated assumptions on the Euler solver. The paper also provides a coercivity analysis of the Newton Jacobian, introduces a blended low/high-order artificial resistivity, and validates the method with a resistive whistler wave convergence study, the GEM reconnection challenge at up to 1024x1024, a new compressible Hall-MHD Orszag-Tang study, and a mesh-topology sensitivity study. I verified the central algebraic identities in (37), the Hall-term cancellation, and the bookkeeping in Appendix B.

Significance. If the results hold, this is a useful contribution: it appears to be among the few structure-preserving schemes for compressible resistive Hall-MHD, with explicit preservation of involution without divergence cleaning, and it ships nontrivial high-resolution GEM and OT benchmarks. The whistler test is a genuine external validation because it uses an independently derived linear dispersion relation. The coercivity analysis is a strength, even though it is not fully exploited. The main uncertainty is the practical nonlinear-solver robustness, not the structural algebra.

major comments (1)
  1. [Sections 3.2, 3.4, 3.5 and Appendix B] Practical well-posedness of the Newton solver and existence of the discrete solution. Proposition 3.2 presupposes a solution (v_h^{n+1}, H_h^{n+1}) of the nonlinear system (32). The coercivity estimates (B.6)-(B.7) give sufficient conditions, but the actual resistivity in Section 3.5, r_low = c_low h_i |v_e|, is O(h) and cannot satisfy the mesh-independent threshold r_min >= (1/2) mu c_e of (B.7); Section 3.4 explicitly states that the chosen resistivity is deliberately not strong enough to guarantee invertibility. The manuscript reports no Newton iteration counts, residual histories, or conditioning estimates, in particular for the 724x724 d_i=0.5 Orszag-Tang run at CFL=0.05. As a result, existence of the discrete solution for the reported parameter regimes is an empirical fact rather than a consequence of the analysis, and the structure-preservation theorem has no computational referen
minor comments (6)
  1. [Proof 3.1, display after (37)] The kinetic-energy terms in the proof are missing the factor rho_i^n. As printed, the proof writes sum_i (1/2) m_i |v_i^{n+1}|^2, whereas the theorem (37) correctly has sum_i m_i (1/2) rho_i^n |v_i^{n+1}|^2. The stated result is correct, but the proof is inconsistent as written.
  2. [Equation (49), Section 3.5] The residual R_h^n contains r^n on the right-hand side even though r^n is the quantity being defined in this section. This makes the definition circular as written; presumably r_h^{n-1} or another explicit previous resistivity is intended.
  3. [Section 4.3, Orszag-Tang setup] The text says 'd_i = 256/10' where the Hall scale is resolved by approximately 10 grid points. This appears to be a typo for d_i = 10/256, consistent with the Figure 4 caption. Please correct.
  4. [Abstract and Appendix A] Minor name/typo issues: 'Crank-Nicholson' should be 'Crank-Nicolson', and 'Nobel-Abel-Stiffened-Gas' should be 'Noble-Abel-Stiffened-Gas'.
  5. [Section 4.2, GEM boundary conditions] The GEM setup uses 'natural boundary conditions' obtained by dropping boundary terms, which is not identical to the H x n = 0 condition used in the energy and involution proofs. State this explicitly so that the theorem/test boundary-condition mismatch is transparent.
  6. [Table 1 and Figure 1, Section 4.1] The convergence rates for momentum components are 1.92-1.94, slightly below the nominal second order achieved for H_x and H_y. The text says 'near-optimal' and this is fair, but the slight discrepancy deserves a one-sentence comment, e.g., a consequence of the split scheme or the quadrature/limiting in the Euler solver.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: structure-preservation claims are proved in-paper from the discretization and explicit solver assumptions, and the whistler-wave benchmark uses an independently derived linear dispersion relation.

full rationale

Proposition 3.2 is a sequential composition of Proposition 3.1 with explicit hypotheses on euler_system_update; the energy, admissibility, entropy, and involution claims are proven directly from equations (32)-(35) and the stated assumptions, not imported from a fit. The whistler-wave test compares the method against the analytic linear dispersion relation (55) and solution (56), which is an external benchmark, so the observed convergence rates 1.92-2.00 are genuine verification rather than a fitted prediction. The artificial resistivity in Section 3.5 is empirically tuned, but it is not renamed as a prediction; the coercivity estimates (B.6)-(B.7) are correctly stated as sufficient conditions, and the paper explicitly disclaims unconditional invertibility: 'we will not use a viscosity strong enough to unconditionally guarantee invertibility' (Section 3.4). The main reliance on prior work [22] supplies the Euler solver and its assumptions; this is a published, separately argued result and does not reduce the Hall/resistive contributions to their own inputs. The absence of Newton-iteration statistics for the hardest runs is a robustness/reproducibility gap, not a circularity. Overall, the derivation chain is self-contained for the claims it makes, with the stated conditional structure made explicit.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The scheme introduces no new physical entities; the artificial resistivity is a numerical regularization whose coefficients are captured as free parameters above. The mathematical claims rest on standard thermodynamic stability assumptions, inherited structural properties of the external Euler solver ([22]), and boundedness hypotheses that the scheme does not itself enforce.

free parameters (4)
  • c_low (low-order artificial resistivity coefficient) = 0.25
    Equation (48): r_low_i = c_low h_i λ_low_i, stated as an 'empirical non-dimensional constant'; used in every simulation, no sensitivity study reported.
  • c_res (residual-based resistivity coefficient) = 1.0
    Equation (52): r_res_i = c_res h_i² R̂_h(x_i); 'in practice we use c_res = 1.0 for all our computations.' Empirical constant, no sensitivity study.
  • CFL constant = 0.5 (0.1 and 0.05 in Section 4.3)
    Standard time-step restriction, hand-adjusted per regime: 'For d_i = 0.25 and 0.5, the CFL constant was lowered to 0.1... A CFL of 0.05 was required to guarantee coercivity of the Jacobian.' The scheme's practical robustness depends on this tuning.
  • Artificial resistivity blend rule = r_i = max{r, min{r_low_i, r_res_i}}
    Section 3.5: an engineered combination of physical, low-order, and residual-based resistivities; the min/max rule localizes the regularization and affects Jacobian conditioning and dissipation.
assumptions (5)
  • domain assumption Thermodynamic stability of the EOS: ∂s/∂e = 1/θ > 0 and concavity of s(v,e) with ∂²s/∂e² ≤ 0, giving ∂θ/∂e ≥ 0 (Appendix A, (A.3)–(A.5)).
    Used in Propositions 2.2 and 3.1 to conclude that Joule heating raises temperature and entropy (eqs. (20)–(21)); satisfied by the ideal-gas EOS (γ = 5/3) used in all numerics.
  • domain assumption The Euler solver euler_system_update satisfies conservation (43), admissibility (44), and the entropy inequality (45) (Section 3.3).
    The full scheme's energy conservation, positivity, and entropy dissipation (Proposition 3.2) are inherited sequentially from this solver; the solver is taken from the authors' prior work [22] and its structural properties are not re-verified numerically in this paper.
  • domain assumption Boundedness hypotheses of Proposition B.1: ‖curlH‖_{L∞} ≤ c_h and ‖(d_i/ρ)curlH − v‖_{L∞} ≤ c_e.
    The coercivity/invertibility guarantees for Newton's Jacobian hold only under these bounds, which the scheme neither enforces nor verifies; in the Hall-dominated Orszag–Tang runs c_e is acknowledged to be one to two orders of magnitude above the ion velocity, outside the regime where the estimates give a practical margin.
  • standard math The linearized whistler-wave dispersion relation (55) with O(r²) accuracy is the exact reference solution for the convergence test (Section 4.1).
    The convergence rates in Table 1 measure deviation from this linearized solution, so the verification inherits the linearization error; with δH = 1e-4 the nonlinear error is small but not quantified.
  • domain assumption Resistivity is fundamental to well-posedness of compressible Hall-MHD (Section 1).
    Motivational premise used to justify studying the resistive model; stated as an assumption ('it may be reasonable to assume that resistivity is fundamental...') and not load-bearing for the numerical method's properties.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A structure-preserving Numerical Method for the Compressible Resistive-Hall-MHD System." pith.science (2026). https://pith.science/paper/VGGDW3GB

@misc{pith2026260714286,
  author       = {Pith},
  title        = {Pith review of: A structure-preserving Numerical Method for the Compressible Resistive-Hall-MHD System},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VGGDW3GB}},
  note         = {Machine review of arXiv:2607.14286}
}
read the original abstract

In this paper, we present a structure-preserving method for the compressible resistive Hall-magnetohydrodynamics (MHD) model. The differential operator is split into two parts: a hydrodynamic part consisting of the compressible Euler equations, and a magnetic part consisting of a system coupling the Lorentz force and the induction equation. The method uses continuous Lagrange elements for the Euler part and a curl-conforming finite element space for the magnetic part. The hydrodynamic part preserves the positivity of the density and internal energy, the conservation of total energy, and the minimum principle for the specific entropy. Owing to the choice of finite elements, the magnetic part preserves the divergence involution constraint. The fluid part is solved using explicit strong-stability-preserving Runge-Kutta (SSP-RK) methods, whereas the magnetic part is solved by Crank-Nicholson method, which requires using Newton's method. Coercivity estimates for the Jacobian of the corresponding Newton iteration are presented. We introduce a high-order artificial resistivity to improve the conditioning of the nonlinear residual and the invertibility of the Jacobian. Several challenging benchmarks, including a smooth whistler wave, the Orszag-Tang vortex for comparing resistive MHD with resistive Hall-MHD, and a magnetic reconnection problem, are solved to validate the robustness and accuracy of the method.

Figures

Figures reproduced from arXiv: 2607.14286 by the authors.

Figure 1
Figure 1. L2 relative error for the whistler wave convergence test [PITH_FULL_IMAGE:figures/full_fig_p022_1.png] view at source ↗
Figure 2
Figure 2. GEM magnetic reconnection challenge out-of-plane component of the current density for [PITH_FULL_IMAGE:figures/full_fig_p024_2.png] view at source ↗
Figure 3
Figure 3. Reconnection rate comparison between resistive MHD with and without the Hall term for [PITH_FULL_IMAGE:figures/full_fig_p025_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Density distribution of the Orszag–Tang vortex on a mesh with [PITH_FULL_IMAGE:figures/full_fig_p026_4.png]
Figure 5
Figure 5. Figure 5: Snapshots of the density and out-of-plane current density for the Orszag–Tang simulation at [PITH_FULL_IMAGE:figures/full_fig_p027_5.png]
Figure 6
Figure 6. Figure 6: Comparison of the GEM magnetic reconnection challenge for three mesh topologies at [PITH_FULL_IMAGE:figures/full_fig_p028_6.png]
Figure 7
Figure 7. Figure 7: Reconnection rate comparison between different mesh topologies for GEM magnetic recon [PITH_FULL_IMAGE:figures/full_fig_p029_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

65 extracted references

  1. [20]

    Lars K. S. Daldorff, Gábor Tóth, Tamas I. Gombosi, Giovanni Lapenta, Jorge Amaya, Stefano Markidis, and Jeremiah U. Brackbill. Two-way coupling of a global Hall magnetohydrodynamics model with a local implicit particle-in-cell model.J. Comput. Phys., 268:236–254, 2014

    2014

  2. [21]

    Lars K. S. Daldorff, Gábor Tóth, Tamas I. Gombosi, Giovanni Lapenta, Jorge Amaya, Stefano Markidis, and Jeremiah U. Brackbill. Two-way coupling of a global Hall magnetohydrodynamics model with a local implicit particle-in-cell model.J. Comput. Phys., 268:236–254, 2014. 35

    2014

  3. [1]

    Ki- netic formulation and global existence for the Hall-Magneto-hydrodynamics system

    Marion Acheritogaray, Pierre Degond, Amic Frouvelle, and Jian-Guo Liu. Ki- netic formulation and global existence for the Hall-Magneto-hydrodynamics system. Kinet. Relat. Models, 4(4):901–918, 2011

    2011

  4. [2]

    Bettina Albers and Krzysztof Wilmanski.Continuum thermodynamics. Part II. Applications and examples, volume 85 ofSeries on Advances in Mathematics for Applied Sciences. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015

    2015

  5. [3]

    Periodic table of the finite elements.Siam News, 47(9):212, 2014

    Douglas N Arnold and Anders Logg. Periodic table of the finite elements.Siam News, 47(9):212, 2014

    2014

  6. [4]

    A semi-implicit Hall-MHD solver using whistler wave preconditioning.Comput

    Lukas Arnold, Jürgen Dreher, and Rainer Grauer. A semi-implicit Hall-MHD solver using whistler wave preconditioning.Comput. Phys. Comm., 178(8):553–557, 2008

    2008

  7. [5]

    Tinsley Oden

    Ivo Babuska and J. Tinsley Oden. Verification and validation in computational engineering and science: basic concepts.Comput. Methods Appl. Mech. Engrg., 193(36-38):4057–4066, 2004

    2004

  8. [6]

    Dinshaw S. Balsara. Self-adjusting, positivity preserving high order schemes for hydrodynamics and magnetohydrodynamics.J. Comput. Phys., 231(22):7504–7517, 2012

    2012

Show all 65 references
  1. [7]

    Balsara, Tobias Rumpf, Michael Dumbser, and Claus-Dieter Munz

    Dinshaw S. Balsara, Tobias Rumpf, Michael Dumbser, and Claus-Dieter Munz. Effi- cient, high accuracy ADER-WENO schemes for hydrodynamics and divergence-free magnetohydrodynamics.J. Comput. Phys., 228(7):2480–2516, 2009. 34

    2009

  2. [8]

    Bard and J

    C. Bard and J. Dorelli. High-performance computational magnetohydrodynamics with python.Computer Physics Communications, 322:110077, 2026

    2026

  3. [9]

    Springer, Cham, 2015

    Sören Bartels.Numerical methods for nonlinear partial differential equations, vol- ume 47 ofSpringer Series in Computational Mathematics. Springer, Cham, 2015

    2015

  4. [10]

    Vanishing viscosity solutions of nonlinear hyperbolic systems.Ann

    Stefano Bianchini and Alberto Bressan. Vanishing viscosity solutions of nonlinear hyperbolic systems.Ann. of Math. (2), 161(1):223–342, 2005

    2005

  5. [11]

    Geospace environmental modeling (gem) magnetic reconnection challenge.Journal of Geophysical Research: Space Physics, 106(A3):3715–3719, 2001

    J Birn, JF Drake, MA Shay, BN Rogers, RE Denton, M Hesse, M Kuznetsova, ZW Ma, A Bhattacharjee, A Otto, et al. Geospace environmental modeling (gem) magnetic reconnection challenge.Journal of Geophysical Research: Space Physics, 106(A3):3715–3719, 2001

    2001

  6. [12]

    Finite element approximation of eigenvalue problems.Acta Numer., 19:1–120, 2010

    Daniele Boffi. Finite element approximation of eigenvalue problems.Acta Numer., 19:1–120, 2010

    2010

  7. [13]

    John wiley & sons, 1991

    Herbert B Callen.Thermodynamics and an Introduction to Thermostatistics. John wiley & sons, 1991

    1991

  8. [14]

    L. Chacón. A scalable multidimensional fully implicit solver for Hall magnetohydro- dynamics.J. Comput. Phys., 526:Paper No. 113789, 20, 2025

    2025

  9. [15]

    Well-posedness for Hall- magnetohydrodynamics.Ann

    Dongho Chae, Pierre Degond, and Jian-Guo Liu. Well-posedness for Hall- magnetohydrodynamics.Ann. Inst. H. Poincaré C Anal. Non Linéaire, 31(3):555– 565, 2014

    2014

  10. [16]

    Singularity formation for the incompressible Hall-MHD equations without resistivity.Ann

    Dongho Chae and Shangkun Weng. Singularity formation for the incompressible Hall-MHD equations without resistivity.Ann. Inst. H. Poincaré C Anal. Non Linéaire, 33(4):1009–1022, 2016

    2016

  11. [17]

    Positivity-preserving DG and central DG methods for ideal MHD equations.J

    Yue Cheng, Fengyan Li, Jianxian Qiu, and Liwei Xu. Positivity-preserving DG and central DG methods for ideal MHD equations.J. Comput. Phys., 238:255–280, 2013

    2013

  12. [18]

    K. N. Chueh, C. C. Conley, and J. A. Smoller. Positively invariant regions for systems of nonlinear diffusion equations.Indiana Univ. Math. J., 26(2):373–392, 1977

    1977

  13. [19]

    StudiesinMath- ematicsanditsApplications, Vol.4.North-HollandPublishingCo., Amsterdam-New York-Oxford, 1978

    PhilippeG.Ciarlet.The finite element method for elliptic problems. StudiesinMath- ematicsanditsApplications, Vol.4.North-HollandPublishingCo., Amsterdam-New York-Oxford, 1978

    1978

  14. [22]

    A structure preserving nu- merical method for the ideal compressible MHD system.J

    Tuan Anh Dao, Murtazo Nazarov, and Ignacio Tomas. A structure preserving nu- merical method for the ideal compressible MHD system.J. Comput. Phys., 508:Pa- per No. 113009, 25, 2024

    2024

  15. [23]

    Dedner, F

    A. Dedner, F. Kemm, D. Kröner, C.-D. Munz, T. Schnitzer, and M. Wesenberg. Hy- perbolic divergence cleaning for the MHD equations.J. Comput. Phys., 175(2):645– 673, 2002

    2002

  16. [24]

    Ideal GLM-MHD: about the entropy consistent nine-wave magnetic field divergence diminishing ideal magnetohydrodynamics equations.J

    DominikDerigs, AndrewR.Winters, GregorJ.Gassner, StefanieWalch, andMarvin Bohm. Ideal GLM-MHD: about the entropy consistent nine-wave magnetic field divergence diminishing ideal magnetohydrodynamics equations.J. Comput. Phys., 364:420–467, 2018

    2018

  17. [25]

    Springer, Cham, [2021] ©2021

    Alexandre Ern and Jean-Luc Guermond.Finite elements I—Approximation and interpolation, volume 72 ofTexts in Applied Mathematics. Springer, Cham, [2021] ©2021

    2021

  18. [26]

    On well-posedness and blow-up for the full compressible Hall-MHD system.Nonlinear Anal

    Jishan Fan, Bashir Ahmad, Tasawar Hayat, and Yong Zhou. On well-posedness and blow-up for the full compressible Hall-MHD system.Nonlinear Anal. Real World Appl., 31:569–579, 2016

    2016

  19. [27]

    On strong solutions to the compressible Hall-magnetohydrodynamic system.Nonlinear Anal

    Jishan Fan, Ahmed Alsaedi, Tasawar Hayat, Gen Nakamura, and Yong Zhou. On strong solutions to the compressible Hall-magnetohydrodynamic system.Nonlinear Anal. Real World Appl., 22:423–434, 2015

    2015

  20. [28]

    Efficient kinetic lattice boltzmann simulation of three-dimensional hall-mhd turbulence.Journal of Plasma Physics, 89(4):905890413, 2023

    Raffaello Foldes, Emmanuel Lévêque, Raffaele Marino, Ermanno Pietropaolo, Alessandro De Rosis, Daniele Telloni, and Fabio Feraco. Efficient kinetic lattice boltzmann simulation of three-dimensional hall-mhd turbulence.Journal of Plasma Physics, 89(4):905890413, 2023

    2023

  21. [29]

    J. P. Freidberg. Ideal magnetohydrodynamic theory of magnetic fusion systems. Rev. Mod. Phys., 54:801–902, Jul 1982

    1982

  22. [30]

    Global existence and optimal decay rates of solutions for compressible Hall-MHD equations.Discrete Contin

    Jincheng Gao and Zheng-An Yao. Global existence and optimal decay rates of solutions for compressible Hall-MHD equations.Discrete Contin. Dyn. Syst., 36(6):3077–3106, 2016

    2016

  23. [31]

    Springer-Verlag, New York, 1996

    Edwige Godlewski and Pierre-Arnaud Raviart.Numerical approximation of hyper- bolic systems of conservation laws, volume 118 ofApplied Mathematical Sciences. Springer-Verlag, New York, 1996

    1996

  24. [32]

    Cambridge university press, 2004

    JP Hans Goedbloed and Stefaan Poedts.Principles of magnetohydrodynamics: with applications to laboratory and astrophysical plasmas. Cambridge university press, 2004

    2004

  25. [33]

    Viscous regularization of the Euler equations and entropy principles.SIAM J

    Jean-Luc Guermond and Bojan Popov. Viscous regularization of the Euler equations and entropy principles.SIAM J. Appl. Math., 74(2):284–305, 2014

    2014

  26. [34]

    Hoelzl, G.T.A

    M. Hoelzl, G.T.A. Huijsmans, S.J.P. Pamela, M. Bécoulet, E. Nardon, F.J. Ar- tola, B. Nkonga, C.V. Atanasiu, V. Bandaru, A. Bhole, D. Bonfiglio, A. Cathey, 36 O. Czarny, A. Dvornova, T. Fehér, A. Fil, E. Franck, S. Futatani, M. Gruca, H. Guil- lard, J.W. Haverkort, I. Holod, D...

    2021

  27. [35]

    Bifurcation analysis of magnetic reconnection in Hall-MHD-systems.Phys

    Holger Homann and Rainer Grauer. Bifurcation analysis of magnetic reconnection in Hall-MHD-systems.Phys. D, 208(1-2):59–72, 2005

    2005

  28. [36]

    Stephen C. Jardin. MHD simulations for fusion applications. InNumerical models for fusion, volume 39/40 ofPanor. Synthèses, pages 177–235. Soc. Math. France, Paris, 2013

    2013

  29. [37]

    On the Cauchy problem for the Hall and electron magnetohydrodynamic equations without resistivity I: Illposedness near degenerate stationary solutions.Ann

    In-Jee Jeong and Sung-Jin Oh. On the Cauchy problem for the Hall and electron magnetohydrodynamic equations without resistivity I: Illposedness near degenerate stationary solutions.Ann. PDE, 8(2):Paper No. 15, 106, 2022

    2022

  30. [38]

    Principles of plasma physics

    Nicholas A Krall, Alvin W Trivelpiece, and KR Symon. Principles of plasma physics. IEEE Transactions on Plasma Science, 2(3):196–196, 1974

    1974

  31. [39]

    Structure-preserving and helicity-conserving finite element approximations and preconditioning for the hall mhd equations.Journal of Computational Physics, 492:112410, 2023

    Fabian Laakmann, Kaibo Hu, and Patrick E Farrell. Structure-preserving and helicity-conserving finite element approximations and preconditioning for the hall mhd equations.Journal of Computational Physics, 492:112410, 2023

    2023

  32. [40]

    On the Cauchy problem of compressible full Hall-MHD equations.Z

    Suhua Lai, Xinying Xu, and Jianwen Zhang. On the Cauchy problem of compressible full Hall-MHD equations.Z. Angew. Math. Phys., 70(5):Paper No. 139, 22, 2019

    2019

  33. [41]

    P. D. Lax. Hyperbolic systems of conservation laws. II.Comm. Pure Appl. Math., 10:537–566, 1957

    1957

  34. [42]

    The noble-abel stiffened-gas equation of state.Physics of Fluids, 28(4):046102, 04 2016

    Olivier Le Métayer and Richard Saurel. The noble-abel stiffened-gas equation of state.Physics of Fluids, 28(4):046102, 04 2016

    2016

  35. [43]

    Lebon, D

    G. Lebon, D. Jou, and J. Casas-Vázquez.Understanding non-equilibrium thermo- dynamics. Springer-Verlag, Berlin, 2008. Foundations, applications, frontiers

    2008

  36. [44]

    Locally divergence-free discontinuous Galerkin methods for MHD equations.J

    Fengyan Li and Chi-Wang Shu. Locally divergence-free discontinuous Galerkin methods for MHD equations.J. Sci. Comput., 22/23:413–442, 2005

    2005

  37. [45]

    M. J. Lighthill. Studies on magneto-hydrodynamic waves and other anisotropic wave motions.Philos. Trans. Roy. Soc. London Ser. A, 252:397–430, 1960

    1960

  38. [46]

    A unified gas kinetic scheme for continuum and rarefied flows v: Multiscale and multi-component plasma transport.Communications in Computational Physics, 22(5):1175–1223, 2017

    Chang Liu and Kun Xu. A unified gas kinetic scheme for continuum and rarefied flows v: Multiscale and multi-component plasma transport.Communications in Computational Physics, 22(5):1175–1223, 2017. 37

    2017

  39. [47]

    Londrillo and L

    P. Londrillo and L. Del Zanna. On the divergence-free condition in Godunov- type schemes for ideal magnetohydrodynamics: the upwind constrained transport method.J. Comput. Phys., 195(1):17–48, 2004

    2004

  40. [48]

    Empirical equations of state for solids

    Ralph Menikoff. Empirical equations of state for solids. InShockWave Science and Technology Reference Library, pages 143–188. Springer, 2007

    2007

  41. [49]

    Ralph Menikoff and Bradley J. Plohr. The riemann problem for fluid flow of real materials.Rev. Mod. Phys., 61:75–130, Jan 1989

    1989

  42. [50]

    Two-and-a-half-dimensional magnetohydrody- namic turbulence.The Physics of Fluids, 25(2):345–349, 1982

    David Montgomery and Leaf Turner. Two-and-a-half-dimensional magnetohydrody- namic turbulence.The Physics of Fluids, 25(2):345–349, 1982

    1982

  43. [51]

    Orszag and Cha-Mei Tang

    Steven A. Orszag and Cha-Mei Tang. Small-scale structure of two-dimensional magnetohydrodynamic turbulence.Journal of Fluid Mechanics, 90(1):129 – 143,

  44. [52]

    Kinetic dissipation and anisotropic heating in a turbulent collisionless plasma.Physics of Plasmas, 16(3), 2009

    TN Parashar, MA Shay, PA Cassak, and WH Matthaeus. Kinetic dissipation and anisotropic heating in a turbulent collisionless plasma.Physics of Plasmas, 16(3), 2009

    2009

  45. [53]

    PL Pritchett. Geospace environment modeling magnetic reconnection challenge: Simulations with a full particle electromagnetic code.Journal of Geophysical Re- search: Space Physics, 106(A3):3783–3798, 2001

    2001

  46. [54]

    A numerical study of magnetic reconnection: a central scheme for Hall MHD

    Xin Qian, Jorge Balbás, Amitava Bhattacharjee, and Hongang Yang. A numerical study of magnetic reconnection: a central scheme for Hall MHD. InHyperbolic problems: theory, numerics and applications, volume 67 ofProc. Sympos. Appl. Math., pages 879–888. Amer. Math. Soc., Provide...

    2009

  47. [55]

    Rossmanith

    James A. Rossmanith. An unstaggered, high-resolution constrained transport method for magnetohydrodynamic flows.SIAM J. Sci. Comput., 28(5):1766–1797, 2006

    2006

  48. [56]

    Alfvénic collisionless magnetic reconnection and the hall term.Journal of Geophysical Research: Space Physics, 106(A3):3759–3772, 2001

    MA Shay, JF Drake, BN Rogers, and RE Denton. Alfvénic collisionless magnetic reconnection and the hall term.Journal of Geophysical Research: Space Physics, 106(A3):3759–3772, 2001

    2001

  49. [57]

    Sitenko and V

    A. Sitenko and V. Malnev.Plasma physics theory, volume 10 ofApplied Mathematics and Mathematical Computation. Chapman & Hall, London, 1995

    1995

  50. [58]

    Small-scale behavior of hall magnetohydro- dynamic turbulence.Physical Review E, 92(6):063102, 2015

    Julia E Stawarz and Annick Pouquet. Small-scale behavior of hall magnetohydro- dynamic turbulence.Physical Review E, 92(6):063102, 2015

    2015

  51. [59]

    Multidimensional Hall magnetohydrody- namics with isotropic or anisotropic thermal pressure: numerical scheme and its validation using solitary waves.J

    Marek Strumik and Krzysztof Stasiewicz. Multidimensional Hall magnetohydrody- namics with isotropic or anisotropic thermal pressure: numerical scheme and its validation using solitary waves.J. Comput. Phys., 330:846–862, 2017

    2017

  52. [60]

    A minimum entropy principle in the gas dynamics equations.Appl

    Eitan Tadmor. A minimum entropy principle in the gas dynamics equations.Appl. Numer. Math., 2(3-5):211–219, 1986. 38

    1986

  53. [61]

    Global existence and exponential sta- bility of solutions for planar compressible Hall-magnetohydrodynamic equations.J

    Qiang Tao, Ying Yang, and Zheng-an Yao. Global existence and exponential sta- bility of solutions for planar compressible Hall-magnetohydrodynamic equations.J. Differential Equations, 263(7):3788–3831, 2017

    2017

  54. [62]

    Torrilhon

    M. Torrilhon. Non-uniform convergence of finite volume schemes for Riemann prob- lems of ideal magnetohydrodynamics.J. Comput. Phys., 192(1):73–94, 2003

    2003

  55. [63]

    Gábor Tóth, Yingjuan Ma, and Tamas I. Gombosi. Hall magnetohydrodynamics on block-adaptive grids.J. Comput. Phys., 227(14):6967–6984, 2008

    2008

  56. [64]

    Positivity-preserving analysis of numerical schemes for ideal magne- tohydrodynamics.SIAM J

    Kailiang Wu. Positivity-preserving analysis of numerical schemes for ideal magne- tohydrodynamics.SIAM J. Numer. Anal., 56(4):2124–2147, 2018

    2018

  57. [65]

    A provably positive discontinuous Galerkin method for multidimensional ideal magnetohydrodynamics.SIAM J

    Kailiang Wu and Chi-Wang Shu. A provably positive discontinuous Galerkin method for multidimensional ideal magnetohydrodynamics.SIAM J. Sci. Comput., 40(5):B1302–B1329, 2018. 39

    2018

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.