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REVIEW 3 major objections 6 minor 51 references

A Kernel Based High Order "Explicit" Unconditionally Stable Constrained Transport Method for Ideal Magnetohydrodynamics

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Pith's one-line read The paper claims that advancing the magnetic vector potential with a kernel-based method of lines transpose makes constrained transport in ideal MHD A-stable, so the diffusion limiter required in the previous 3D scheme can be removed…

desk verdict A genuinely useful extension of the authors' kernel-based MOLT to vector-potential CT, with a clean blast-wave comparison—but the 'unconditionally stable in 3D' claim outruns the 1D scalar proof and the evidence is all at CFL 0.5. read the letter →

arxiv 1908.01023 v1 pith:CLN2MLTA submitted 2019-08-02 math.NA cs.NA

Firat Cakir , Andrew Christlieb , Yan Jiang This is my paper · ORCID
classification math.NAcs.NA MSC 65M0665M1265M2076W0535L65
keywords idealmagnetohydrodynamicsconstrainedtransportmagneticvectorpotentialmethodoflinestransposekernel-basedschemeA-stabilitydivergence-freeconditionWENO
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 claims that the magnetic vector potential equation in ideal magnetohydrodynamics can be advanced by a kernel-based method of lines transpose that is A-stable, so the time step of the potential update is not limited by the usual explicit stability condition. The payoff is that the artificial diffusion limiter used in the authors' earlier 3D constrained-transport scheme is no longer needed for stability. Recovering the magnetic field as the 4th-order central curl of the potential makes the discrete divergence identically zero in both 2D and 3D. A sympathetic reader should care because, if the claim holds, divergence-free MHD simulations with strong shocks can be run with a mesh-aligned, AMR-friendly update that does not smear shock structure through added resistivity.

What carries the argument

The load-bearing object is the kernel-based approximation of the one-sided derivatives $A_x^-$ and $A_x^+$, built from the operators $L_L = I + \frac{1}{\alpha}\partial_x$ and $L_R = I - \frac{1}{\alpha}\partial_x$. With $\alpha = \beta/(c\,\Delta t)$, the inverse operators become successive convolution integrals, and the derivative is expressed as the series $\frac{1}{\alpha}\partial_x = \sum_{p\ge1} D_L^p$; truncating the series gives $k$th-order accuracy in $\Delta t$, and for $0 < \beta \le \beta_{k,\max}$ the 1D scalar advection scheme is A-stable. WENO-based quadrature of the convolution integrals supplies the nonlinear weights and filters that control oscillations at shocks. This machinery defines the numerical Hamiltonian in the Lax-Friedrichs flux, and is what allows the potential update to remain stable with large time steps.

What would settle it

Take the 3D potential update alone with a non-grid-aligned constant velocity such as $u=(1,1,1)$, periodic boundary conditions, a smooth random initial $A$, and a time step several times the explicit CFL limit with $\beta$ chosen as in the paper; monitor the Fourier energy of $A$ over many steps. If the energy grows without bound while the 1D scalar test remains bounded, the transfer of A-stability to the 3D system fails.

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Extended reading notes

Core claim

The central claim is that replacing local derivative approximations in the magnetic-vector-potential update with global convolution-kernel operators turns explicit Runge-Kutta time stepping into an unconditionally stable scheme for the potential equation. In 2D the potential is a scalar advection equation; in 3D it is the weakly hyperbolic system $\partial_t A + N_1 A_x + N_2 A_y + N_3 A_z = 0$ obtained with the Weyl gauge. The authors apply the 1D kernel derivative operators direction-by-direction with Lax-Friedrichs flux splitting, halve the 1D stability bound on $\beta$ in 2D to maintain unconditional stability, and claim the same idea transfers to the coupled 3D system. The magnetic field is then recovered as $B = \nabla \times A$ with 4th-order central differences, which cancels in the discrete divergence by commutativity of the difference operators. The paper's key numerical comparison is the 3D blast wave, where removing the diffusion limiter raises the peak $\|u\|$ from 261 to 320.

Load-bearing premise

The scheme's stability guarantee was proved only for a one-dimensional scalar advection equation, and the paper assumes the same property carries over to the three-dimensional coupled magnetic-potential system with variable coefficients, WENO filters, and flux splitting; no separate proof is given for the system.

Editorial extensions

If this is right

  • The 3D constrained-transport update no longer needs the artificial resistivity term that stabilized the earlier scheme, so strong-shock solutions are less diffused; in the 3D blast wave the peak $\|u\|$ increases from 261 to 320.
  • Because the corrected field is $B = \nabla \times A$ with central differences, the discrete divergence $\nabla \cdot B$ vanishes by commutativity of difference operators, independent of the accuracy of $A$.
  • The kernel-based potential update is mesh-aligned and does not require solving a Poisson equation, so it fits an adaptive mesh refinement framework.
  • The A-stable potential update means the time step is governed by the base MHD solver rather than by the divergence-correction step.

Reading between the lines

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

  • The paper proves A-stability for the 1D scalar advection equation; the 3D weakly hyperbolic system is treated by analogy, so a numerical amplification-factor study of the full 3D system would either confirm the transfer or reveal a hidden step-size restriction.
  • If the transfer holds, the same kernel-based update could be applied to other curl-form evolution equations where divergence constraints matter, such as vorticity-streamfunction formulations or reduced MHD models.
  • The blast-wave comparison suggests that diffusion limiters used purely for numerical stability can alter shock physics; a limiter-on/limiter-off comparison in other MHD codes would be a cheap diagnostic.
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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

3 major / 6 minor

Summary. The paper proposes a constrained-transport scheme for ideal MHD in which the magnetic vector potential is advanced by a kernel-based method-of-lines-transpose derivative solver, and the magnetic field is corrected as a discrete curl of the potential. The authors claim that this potential update is A-stable without artificial resistivity, unlike their earlier CT scheme [21], and that the discrete curl identity preserves the divergence-free condition on the mesh. The numerical section includes 2D and 3D tests: a smooth vortex accuracy study, Orszag-Tang vortex, cloud shock, blast wave, and field loop problems, together with a comparison in Section 6.8 that attributes differences in the 3D blast wave to the removed diffusion limiter.

Significance. If the stability claim holds, the contribution is significant: an AMR-friendly, mesh-aligned, unstaggered CT method that avoids diffusion limiters would remove a known limitation of the authors' earlier scheme and improve robustness for strong-shock MHD. The paper deserves credit for grounding the scalar derivative approximations in prior proven results (Theorems 4.1 and 4.2), for the discrete divergence-free identity in Section 5.2, and for the Section 6.8 control experiment that directly tests the diffusion-limiter hypothesis. The smooth vortex study (Table 6.2) shows high-order convergence of the coupled scheme. However, the headline 3D no-diffusion-limiter claim currently rests on an unproven transfer of 1D scalar A-stability to a weakly hyperbolic 3D system, and all reported tests are run at CFL 0.5, so the unconditional-stability assertion is not yet established.

major comments (3)
  1. [Section 4.6, Eq. (4.42)] The central claim that the 3D potential update is A-stable and therefore needs no diffusion limiter is not proven. Theorem 4.1 establishes A-stability only for the 1D scalar linear advection equation with periodic boundary conditions, while the 3D update is applied to the weakly hyperbolic system (2.12), whose flux Jacobian (2.13) has incomplete eigenvectors in four directions (Section 2.2), with velocity-dependent coefficients, a componentwise Lax-Friedrichs splitting (4.42a)-(4.42c), WENO-filtered derivative approximations, and, in several test problems, outflow boundary conditions. Section 4.6's statement that 'we can simply apply the ideas from the 2D case' is not a stability argument, and no eigenvalue, energy, or normal-mode analysis of the coupled update is given. A linearized von Neumann analysis for frozen coefficients, at minimum, is needed to support the no-diffusion-limiter claim.
  2. [Section 4.5 and Eq. (6.1)] The 2D statement that beta_max 'need[s] to be chosen as half of that for the 1D case to ensure the unconditional stability' is asserted without derivation, and no analogous 3D beta_max is given in Section 4.6. The beta values actually used in the Section 6 tests are not reported, so the claimed stability guarantee is not reproducible. Furthermore, every numerical experiment uses CFL=0.5 (Eq. (6.1)), which is within the reach of ordinary explicit schemes and cannot distinguish an unconditionally stable update from a conditionally stable one. Please provide large-time-step tests (e.g., CFL 2, 5, 10) for a 2D and a 3D field-loop problem, or a computed stability-region diagram, and report the beta values used.
  3. [Section 7 and Sections 6.7-6.8] The conclusion states that 'the new method is unconditionally stable,' but the coupled base MHD solver remains explicit and is advanced with CFL=0.5 (Eq. (6.1)). If the unconditional-stability claim applies only to the potential update and not to the full MHD scheme, the conclusion should be scoped accordingly; as written, the claim overstates what the discrete equations and tests establish. The Section 6.7-6.8 comparison (maximum ||u|| of 261, 320, and 265 with the limiter) is informative for the diffusion-limiter hypothesis, but it does not by itself establish the stability of the 3D update.
minor comments (6)
  1. [Section 6.3] The initial A3 formula for x greater than or equal to 0.05 is printed as -0.56418958(x-0.005); the two branches should match at x=0.05, so the subtraction point is likely a typo for 0.05.
  2. [Section 6.1] The initial data vector lists (rho,u1,u3,u3,p,...), which omits u2 and repeats u3; it should presumably read (rho,u1,u2,u3,p,...).
  3. [Section 6.2] The initial density is written as gamma2; this should be gamma^2 (or otherwise defined), and the initial data line would be clearer if commas were used consistently.
  4. [Table 6.2] The reported L1 convergence rates at 160x160 and 320x320 (4.078 and 3.924) are above the claimed third-order design; a sentence explaining the observed rates would be helpful.
  5. [Section 6.6, Fig. 6.6] The caption says '128x128 grid points' for the 3D field loop, but the computation is on a 128x128x128 mesh; the figure caption should state the full mesh size.
  6. [Section 5.2] No quantitative measure of the discrete divergence error (e.g., ||D dot B||_infinity over time) is reported; the pointwise identity (5.4) is reassuring, but a time-history plot would demonstrate that the correction step performs as intended in practice.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation found: the A-stability claim is inherited from the authors' prior work and then extrapolated to 3D, which is a missing-proof/correctness concern rather than a circular reduction.

full rationale

The derivation chain does not reduce to its own inputs. The kernel-based derivative operators and the A-stability theorem are taken from the authors' own prior papers [17, 18]; those are genuine prior results, proved there for a 1D scalar linear advection equation under stated assumptions, not assumptions manufactured in this paper to force the advertised conclusion. The 2D magnetic-potential equation (4.34) is a scalar advection equation that directly fits that framework, and the 3D update (4.42a-c) is a componentwise Lax-Friedrichs splitting of the weakly hyperbolic system (2.12). No parameter is fitted to produce the 'no diffusion limiter' outcome; Section 6.8 directly tests that claim by re-adding the old diffusion limiter and showing the solution reverts to the previous method's behavior, which is a genuine empirical check. The divergence-free property in Section 5 is enforced by construction because the corrected field is defined as a discrete curl of A with commuting differences, so this is an honest design property of constrained transport rather than a fitted prediction or a renamed result. The main weakness is that Section 4.6 asserts 'we can simply apply the ideas from the 2D case' and Section 4.5 halves beta_max 'to ensure unconditional stability' without a stability proof for the coupled 3D weakly hyperbolic system; Theorem 4.1 does not cover that setting. This is an extrapolation and an omitted proof, and it should be flagged as a correctness/rigor risk, not as a circular argument. Because the central stability premise is borrowed from self-citations but is not circularly deduced, the appropriate circularity score is low.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the kernel-based derivative approximation theory from the authors' earlier work [17,18], on the weakly hyperbolic vector potential formulation, and on the Weyl gauge. The only tunable method parameter reported is beta, which is not given numerically. No new physical entities are introduced.

free parameters (1)
  • beta (stability parameter in alpha = beta/(c*dt)) = not reported (must be <= beta_k,max from Table 4.1)
    Controls the kernel decay and the claimed A-stability. The paper never states the value used in the numerical tests, hurting reproducibility.
assumptions (5)
  • standard math Kernel-based derivative approximations (4.11)/(4.16) are kth-order accurate and A-stable for linear scalar advection with alpha = beta/(c*dt) (Theorems 4.1 and 4.2 from refs [17,18]).
    Proved in the authors' earlier papers and cited, not re-proven here. The paper relies on it to claim unconditional stability for the vector potential updates.
  • domain assumption Stability of the 3D weakly hyperbolic vector potential system (2.12) follows from the scalar 1D kernel analysis without additional proof.
    Section 4.6 applies the scalar HJ machinery directly to the coupled system; no eigenvalue or energy analysis is given for the system.
  • domain assumption The Weyl gauge, psi = 0, yields the correct and stable potential evolution equation (2.11).
    Adopted from Helzel et al. [31]; the paper does not analyze gauge choice effects on the discrete solution.
  • domain assumption WENO quadrature and nonlinear filter sigma control oscillations while preserving order for discontinuous derivatives of A.
    Borrowed from the authors' kernel HJ paper [17]; no convergence analysis for nonsmooth A is provided here.
  • domain assumption Positivity-preserving limiter from [21] prevents negative density or pressure in blast wave tests without affecting the CT update.
    Applied in Sections 6.4 and 6.7 with no analysis of its interaction with the new kernel scheme.

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Pith. "Pith review of A Kernel Based High Order "Explicit" Unconditionally Stable Constrained Transport Method for Ideal Magnetohydrodynamics." pith.science (2026). https://pith.science/paper/CLN2MLTA

@misc{pith2026190801023,
  author       = {Pith},
  title        = {Pith review of: A Kernel Based High Order "Explicit" Unconditionally Stable Constrained Transport Method for Ideal Magnetohydrodynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CLN2MLTA}},
  note         = {Machine review of arXiv:1908.01023}
}
abstract

The ideal Magnetohydrodynamics (MHD) equations are challenging because one needs to maintain the divergence free condition, $\nabla \cdot \Bv = 0$. Many numerical methods have been developed to enforce this condition. In this work, we further our work on mesh aligned constrained transport by developing a new kernel based approach for the vector potential in 2D and 3D. The approach for solving the vector potential is based on the method of lines transpose and is A-stable, eliminating the need for diffusion limiters needed in our previous work in 3D. The work presented here is an improvement over the previous method in the context of problems with strong shocks due to the fact that we could eliminate the diffusion limiter that was needed in our previous version of constrained transport. The method is robust and has been tested on the 2D and 3D cloud shock, blast wave and field loop problems.

Figures

Figures reproduced from arXiv: 1908.01023 by the authors.

Figure 4.1
Figure 4.1. The structure of the stencils in WENO integration. with the linear weights dr satisfying P2 r=0 dr = 1. 3. We develop the following nonlinear weights ωr using the linear weights dr ωr = ˜ωr/ X 2 s=0 ω˜s, r = 0, 1, 2, (4.27) with ω˜r = dr Å 1 + τ5  + βr ã . We take  = 10−6 as a small positive number,  > 0, in our numerical test problems to avoid zero at the denominator. The smoothness indicator βr is determined as… view at source ↗
Figure 6.2
Figure 6.2. Orszag-Tang vortex problem. Contour plots of density at t = 3 with 192 × 192 grid points [PITH_FULL_IMAGE:figures/full_fig_p025_6_2.png] view at source ↗
Figure 6.3
Figure 6.3. Cloud shock problem. Contour plots of kBk at t = 0.06 with 512 × 512 grid points. We use a domain [−0.5, 0.5] × [−0.5, 0.5] with 256 × 256 mesh. Outflow boundary conditions are applied everywhere. The results are shown in [PITH_FULL_IMAGE:figures/full_fig_p026_6_3.png] view at source ↗
Figures from the paper (7 more)
Figure 6.5
Figure 6.5. Figure 6.5: We observe that the solution maintains the circular symmetry of the initial condition. [PITH_FULL_IMAGE:figures/full_fig_p026_6_5.png]
Figure 6.4
Figure 6.4. Figure 6.4: 2D Blast wave problem. Contour plots at time t = 0.01 with 256 × 256 grid points. (a) density; (b) pressure; (c) the norm of u; (d) magnetic pressure kBk. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_6_4.png]
Figure 6.5
Figure 6.5. Figure 6.5: 2D Field loop. Contour plots at time t = 1 with 128 × 128 grid points. a) Magnetic Pressure kBk; b) Magnetic Potential A3 . 6.6 3D Field Loop We tested an advecting field loop which moved diagonally across the boundary with an arbitrary initial angle. The initial con…
Figure 6.6
Figure 6.6. Figure 6.6: 3D Field loop. Contour plots at time t = 1 with 128 × 128 grid points. The loop has been advected around the grid once. a) Magnetic Pressure; b) Magnetic Potential. 6.7 3D Blast wave In this section we investigate the 3D version of the blast wave problem to show the …
Figure 6.7
Figure 6.7. Figure 6.7: 3D Blast wave problem. Contour plots at time t = 0.01 with 150 × 150 grid points 30 [PITH_FULL_IMAGE:figures/full_fig_p030_6_7.png]
Figure 6.8
Figure 6.8. Figure 6.8: 3D Blast wave problem. Contour plots at time t = 0.01 with 150 × 150 grid points. While the max kuk value for previous method is 261, it is 320 for the kernel-based method. 32 [PITH_FULL_IMAGE:figures/full_fig_p032_6_8.png]
Figure 6.9
Figure 6.9. Figure 6.9: 3D Blast wave with diffusion. The max kuk value is 265. blast wave problems. 33 [PITH_FULL_IMAGE:figures/full_fig_p033_6_9.png]

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Works this paper leans on

51 extracted references · 51 canonical work pages

  1. [21]

    A. J. Christlieb, J. A. Rossmanith, and Q. Tang. Finite difference weighted essentially non- oscillatory schemes with constrained transport for ideal magnetohydrodynamics. Journal of Computational Physics, 268:302–325, 2014

    work page 2014

  2. [1]

    D. S. Balsara. Divergence-free adaptive mesh refinement for magnetohydrodynamics. Journal of Computational Physics , 174(2):614–648, 2001

    work page 2001

  3. [2]

    D. S. Balsara. Second-order-accurate schemes for magnetohydrodynamics with divergence-free reconstruction. The Astrophysical Journal Supplement Series , 151(1):149, 2004

    work page 2004

  4. [3]

    D. S. Balsara. Divergence-free reconstruction of magnetic fields and weno schemes for magne- tohydrodynamics. Journal of Computational Physics , 228(14):5040–5056, 2009

    work page 2009

  5. [4]

    D. S. Balsara and J. Kim. A comparison between divergence-cleaning and staggered-mesh formulations for numerical magnetohydrodynamics. The Astrophysical Journal , 602(2):1079, 2004

    work page 2004

  6. [5]

    D. S. Balsara, C. Meyer, M. Dumbser, H. Du, and Z. Xu. Efficient implementation of ader schemes for euler and magnetohydrodynamical flows on structured meshes–speed comparisons with runge–kutta methods. Journal of Computational Physics , 235:934–969, 2013

    work page 2013

  7. [6]

    D. S. Balsara, T. Rumpf, M. Dumbser, and C.-D. Munz. Efficient, high accuracy ader-weno schemes for hydrodynamics and divergence-free magnetohydrodynamics. Journal of Compu- tational Physics, 228(7):2480–2516, 2009

    work page 2009

  8. [7]

    D. S. Balsara and D. S. Spicer. A staggered mesh algorithm using high order godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamic simulations. Journal of Compu- tational Physics, 149(2):270–292, 1999

    work page 1999

Show all 51 references
  1. [8]

    Borges, M

    R. Borges, M. Carmona, B. Costa, and W. S. Don. An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws. Journal of Computational Physics , 227(6):3191–3211, 2008

    2008

  2. [9]

    J. U. Brackbill and D. C. Barnes. The effect of nonzero? · b on the numerical solution of the magnetohydrodynamic equations. Journal of Computational Physics , 35(3):426–430, 1980

    1980

  3. [10]

    Causley, H

    M. Causley, H. Cho, and A. Christlieb. Method of lines transpose: Energy gradient flows using direct operator inversion for phase-field models. SIAM Journal on Scientific Computing , 39(5):B968–B992, 2017. 34

    2017

  4. [11]

    Causley, A

    M. Causley, A. Christlieb, B. Ong, and L. Van Groningen. Method of lines transpose: An implicit solution to the wave equation. Mathematics of Computation, 83(290):2763–2786, 2014

    2014

  5. [12]

    M. F. Causley, H. Cho, A. J. Christlieb, and D. C. Seal. Method of lines transpose: High order L-stableO(N) schemes for parabolic equations using successive convolution. SIAM Journal on Numerical Analysis, 54(3):1635–1652, 2016

    2016

  6. [13]

    M. F. Causley and A. J. Christlieb. Higher order a-stable schemes for the wave equation using a successive convolution approach. SIAM Journal on Numerical Analysis, 52(1):220–235, 2014

    2014

  7. [14]

    M. F. Causley, A. J. Christlieb, Y. Guclu, and E. Wolf. Method of lines transpose: A fast implicit wave propagator. arXiv preprint arXiv:1306.6902 , 2013

    2013 arXiv

  8. [15]

    Cheng, F

    Y. Cheng, F. Li, J. Qiu, and L. Xu. Positivity-preserving dg and central dg methods for ideal mhd equations. Journal of Computational Physics , 238:255–280, 2013

    2013

  9. [16]

    Christlieb, W

    A. Christlieb, W. Guo, and Y. Jiang. A WENO-based Method of Lines Transpose approach for Vlasov simulations. Journal of Computational Physics , 327:337–367, 2016

    2016

  10. [17]

    explicit

    A. Christlieb, W. Guo, and Y. Jiang. Kernel based high order “explicit” unconditionally- stable scheme for nonlinear degenerate advection-diffusion equations. arXiv preprint arXiv:1707.09294, 2017

    2017 arXiv

  11. [18]

    explicit

    A. Christlieb, W. Guo, and Y. Jiang. A kernel based high order “explicit” unconditionally sta- ble scheme for time dependent hamilton–jacobi equations. Journal of Computational Physics , 379:214–236, 2019

    2019

  12. [19]

    A. J. Christlieb, X. Feng, D. C. Seal, and Q. Tang. A high-order positivity-preserving single- stage single-step method for the ideal magnetohydrodynamic equations. Journal of Computa- tional Physics, 316:218–242, 2016

    2016

  13. [20]

    A. J. Christlieb, Y. Liu, Q. Tang, and Z. Xu. Positivity-preserving finite difference weighted eno schemes with constrained transport for ideal magnetohydrodynamic equations. SIAM Journal on Scientific Computing , 37(4):A1825–A1845, 2015. 35

    2015

  14. [22]

    Dai and P

    W. Dai and P. R. Woodward. A simple finite difference scheme for multidimensional magne- tohydrodynamical equations. Journal of Computational Physics , 142(2):331–369, 1998

    1998

  15. [23]

    De Sterck

    H. De Sterck. Multi-dimensional upwind constrained transport on unstructured grids for ’shallow water’ magnetohydrodynamics. In AIAA Computational Fluid Dynamics Conference, 15 th, Anaheim, CA , 2001

    2001

  16. [24]

    Dedner, F

    A. Dedner, F. Kemm, D. Kr¨ oner, C.-D. Munz, T. Schnitzer, and M. Wesenberg. Hyperbolic divergence cleaning for the mhd equations. Journal of Computational Physics , 175(2):645–673, 2002

    2002

  17. [25]

    C. R. DeVore. Flux-corrected transport techniques for multidimensional compressible magne- tohydrodynamics. Journal of Computational Physics , 92(1):142–160, 1991

    1991

  18. [26]

    E. Dorfi. Numerical methods for astrophysical plasmas. Computer Physics Communications , 43(1):1–15, 1986

    1986

  19. [27]

    C. R. Evans and J. F. Hawley. Simulation of magnetohydrodynamic flows-a constrained trans- port method. The Astrophysical Journal, 332:659–677, 1988

    1988

  20. [28]

    Fey and M

    M. Fey and M. Torrilhon. A constrained transport upwind scheme for divergence-free advec- tion. In Hyperbolic problems: theory, numerics, applications , pages 529–538. Springer, 2003

    2003

  21. [29]

    P. Fu, F. Li, and Y. Xu. Globally divergence-free discontinuous galerkin methods for ideal magnetohydrodynamic equations. Journal of Scientific Computing , 77(3):1621–1659, 2018

    2018

  22. [30]

    Gottlieb, C.-W

    S. Gottlieb, C.-W. Shu, and E. Tadmor. Strong stability-preserving high-order time discretiza- tion methods. SIAM review, 43(1):89–112, 2001

    2001

  23. [31]

    Helzel, J

    C. Helzel, J. A. Rossmanith, and B. Taetz. An unstaggered constrained transport method for the 3d ideal magnetohydrodynamic equations. Journal of Computational Physics , 230(10):3803–3829, 2011. 36

    2011

  24. [32]

    Helzel, J

    C. Helzel, J. A. Rossmanith, and B. Taetz. A high-order unstaggered constrained-transport method for the three-dimensional ideal magnetohydrodynamic equations based on the method of lines. SIAM Journal on Scientific Computing , 35(2):A623–A651, 2013

    2013

  25. [33]

    Jiang and D

    G.-S. Jiang and D. Peng. Weighted ENO schemes for Hamilton–Jacobi equations. SIAM Journal on Scientific computing , 21(6):2126–2143, 2000

    2000

  26. [34]

    Jiang and C.-W

    G.-S. Jiang and C.-W. Shu. Efficient implementation of weighted eno schemes. Journal of computational physics, 126(1):202–228, 1996

    1996

  27. [35]

    S. Kawai. Divergence-free-preserving high-order schemes for magnetohydrodynamics: An ar- tificial magnetic resistivity method. Journal of Computational Physics , 251:292–318, 2013

    2013

  28. [36]

    R. J. LeVeque. Wave propagation algorithms for multidimensional hyperbolic systems. Journal of Computational Physics , 131(2):327–353, 1997

    1997

  29. [37]

    F. Li, L. Xu, and S. Yakovlev. Central discontinuous galerkin methods for ideal mhd equa- tions with the exactly divergence-free magnetic field. Journal of Computational Physics , 230(12):4828–4847, 2011

    2011

  30. [38]

    Londrillo and L

    P. Londrillo and L. Del Zanna. High-order upwind schemes for multidimensional magnetohy- drodynamics. The Astrophysical Journal, 530(1):508, 2000

    2000

  31. [39]

    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. Journal of Compu- tational Physics, 195(1):17–48, 2004

    2004

  32. [40]

    G. K. Parks. Physics of space plasmas-an introduction. Redwood City, CA, Addison-Wesley Publishing Co., 1991, 547 p. , 1991

    1991

  33. [41]

    K. G. Powell. An approximate riemann solver for magnetohydrodynamics (that works more than one dimension). Technical report, INSTITUTE FOR COMPUTER APPLICATIONS IN SCIENCE AND ENGINEERING HAMPTON VA, 1994

    1994

  34. [42]

    K. G. Powell, P. L. Roe, T. J. Linde, T. I. Gombosi, and D. L. De Zeeuw. A solution- adaptive upwind scheme for ideal magnetohydrodynamics. Journal of Computational Physics , 154(2):284–309, 1999. 37

    1999

  35. [43]

    J. A. Rossmanith. An unstaggered, high-resolution constrained transport method for magne- tohydrodynamic flows. SIAM Journal on Scientific Computing , 28(5):1766–1797, 2006

    2006

  36. [44]

    D. Ryu, F. Miniati, T. Jones, and A. Frank. A divergence-free upwind code for multidimen- sional magnetohydrodynamic flows. The Astrophysical Journal, 509(1):244, 1998

    1998

  37. [45]

    D. C. Seal, Q. Tang, Z. Xu, and A. J. Christlieb. An explicit high-order single-stage single- step positivity-preserving finite difference weno method for the compressible euler equations. Journal of Scientific Computing , 68(1):171–190, 2016

    2016

  38. [46]

    C.-W. Shu. High order weighted essentially nonoscillatory schemes for convection dominated problems. SIAM review, 51(1):82–126, 2009

    2009

  39. [47]

    Torrilhon

    M. Torrilhon. Locally divergence-preserving upwind finite volume schemes for magnetohydro- dynamic equations. SIAM Journal on Scientific Computing , 26(4):1166–1191, 2005

    2005

  40. [48]

    G. T´ oth. The∇· B = 0 constraint in shock-capturing magnetohydrodynamics codes. Journal of Computational Physics , 161(2):605–652, 2000

    2000

  41. [49]

    J. R. Wilson. Some magnetic effects in stellar collapse and accretion fn1. Annals of the New York Academy of Sciences, 262(1):123–132, 1975

    1975

  42. [50]

    K. Yee. Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media. IEEE Transactions on antennas and propagation , 14(3):302–307, 1966

    1966

  43. [51]

    A. L. Zachary, A. Malagoli, and P. Colella. A higher-order godunov method for multidimen- sional ideal magnetohydrodynamics. SIAM Journal on Scientific Computing , 15(2):263–284, 1994. 38

    1994

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