Shock-Capturing Approach and Nonevolutionary Solutions in Magnetohydrodynamics

Barmin, A. A.; Kulikovskiy, A. G.; Pogorelov, N. V.

doi:10.1006/jcph.1996.0121

Cited by 89 publications

(73 citation statements)

References 9 publications

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“…This is the combination of an intermediate shock and a rarefaction wave, a feature sometimes encountered in coplanar problems due to the non-strict hyperbolicity of MHD. Given its nature, it cannot be found by exact Riemann solvers, and the physical acceptability itself as a solution of the ideal MHD equations is still debated (Barmin et al 1996;Myong & Roe 1998;Torrilhon 2004). On the other hand, this feature is invariably found by means of any numerical scheme, where some sort of dissipation, either physical or numerical, is always present.…”

Section: Shock Tube With Gauge Effectsmentioning

confidence: 99%

ECHO: a Eulerian conservative high-order scheme for general relativistic magnetohydrodynamics and magnetodynamics

Zanna¹,

Zanotti²,

Bucciantini

et al. 2007

A&A

318

448

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Aims. We present a new numerical code, ECHO, based on a Eulerian conservative high-order scheme for time dependent threedimensional general relativistic magnetohydrodynamics (GRMHD) and magnetodynamics (GRMD). ECHO is aimed at providing a shock-capturing conservative method able to work at an arbitrary level of formal accuracy (for smooth flows), where the other existing GRMHD and GRMD schemes yield an overall second order at most. Moreover, our goal is to present a general framework based on the 3 + 1 Eulerian formalism, allowing for different sets of equations and different algorithms and working in a generic space-time metric, so that ECHO may be easily coupled to any solver for Einstein's equations. Methods. Our finite-difference conservative scheme previously developed for special relativistic hydrodynamics and MHD is extended here to the general relativistic case. Various high-order reconstruction methods are implemented and a two-wave approximate Riemann solver is used. The induction equation is treated by adopting the upwind constrained transport (UCT) procedures, appropriate to preserving the divergence-free condition of the magnetic field in shock-capturing methods. The limiting case of magnetodynamics (also known as force-free degenerate electrodynamics) is implemented by simply replacing the fluid velocity with the electromagnetic drift velocity and by neglecting the contribution of matter to the stress tensor. Results. ECHO is particularly accurate, efficient, versatile, and robust. It has been tested against several astrophysical applications, like magnetized accretion onto black holes and constant angular momentum thick disks threaded by toroidal fields. A novel test of the propagation of large-amplitude, circularly polarized Alfvén waves is proposed, and this allows us to prove the spatial and temporal high-order properties of ECHO very accurately. In particular, we show that reconstruction based on a monotonicity-preserving (MP) filter applied to a fixed 5-point stencil gives highly accurate results for smooth solutions, both in flat and curved metric (up to the nominal fifth order), while at the same time providing sharp profiles in tests involving discontinuities.

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Section: Shock Tube With Gauge Effectsmentioning

confidence: 99%

ECHO: a Eulerian conservative high-order scheme for general relativistic magnetohydrodynamics and magnetodynamics

Zanna¹,

Zanotti²,

Bucciantini

et al. 2007

A&A

318

448

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“…In the ideal MHD model, the finitevolume method is used, and the numerical flux is evaluated by the total variation diminishing (TVD) Lax-Friedrich scheme. 13) In order to achieve third-order accuracy in space and second-order accuracy in time, monotone upstreamcentered schemes conservation laws (MUSCL) interpolation and second-order Rung-Kutta time integration are used. In addition, with a view to realize the TVD condition, a minimal model (MINMOD) limiter is used.…”

Section: Simulation Codementioning

confidence: 99%

Quantitative Evaluation of Ion Kinetic Effect in Magnetic Field Inflation by the Injection of a Plasma Jet

Kajimura

Funaki

Nishida

et al. 2011

Trans. Japan Soc. Aero. S Sci.

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The steady state of dipolar magnetic field expansion is examined by injecting a plasma jet from the center of the dipolar magnetic field (magnetic inflation). Using a two-dimensional hybrid particle-in-cell (PIC) code taking into account the finite Larmor radius effect, we examine the magnetic field inflation process when Argon (Ar) plasma is injected into a dipole magnetic field generated by a simple hoop coil; the plasma is injected within an angle of 30 in the polar direction. Compared to ideal magneto hydrodynamics (MHD) results, the results obtained using the hybrid PIC code are more accurate since the finite ion Larmor radius effect decreases the flow of magnetic flux with respect to the flow of the plasma jet.

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“…They also demonstrated numerically that the intermediate shock does not emerge if coplanarity of the solution is broken by inserting a thin layer within which the magnetic field rotates continuously. Barmin et al (1996), on the other hand, found numerically that the compound wave breaks into a rotational discontinuity and a slow shock if the exact coplanarity is perturbed. Although Falle & Komissarov (2001) agreed with Wu (1990) that the temporary survival of 1 → 3, 2 → 4 and 1 → 4 shocks in their interactions with Alfvén waves is due to the non-unique shock structures, it was claimed that they should be regarded as transients.…”

Section: Introductionmentioning

confidence: 99%

Regular and non-regular solutions of the Riemann problem in ideal magnetohydrodynamics

Takahashi¹,

Yamada

2012

J. Plasma Phys.

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We have built a code to numerically solve the Riemann problem in ideal magnetohydrodynamics (MHD) for an arbitrary initial condition to investigate a variety of solutions more thoroughly. The code can handle not only regular solutions, in which no intermediate shocks are involved, but also all types of non-regular solutions if any. As a first application, we explored the neighborhood of the initial condition that was first picked up by Brio & Wu (1988) and has been frequently employed in the literature as a standard problem to validate numerical codes. Contrary to the conventional wisdom that there will always be a regular solution, we found an initial condition, for which there is no regular solution but a non-regular one. The latter solution has only regular solutions in its neighborhood and actually sits on the boundary of regular solutions. This implies that the regular solutions are not sufficient to solve the ideal MHD Riemann problem and suggests that at least some types of non-regular solutions are physical. We also demonstrate that the non-regular solutions are not unique. In fact, we found for the Brio & Wu initial condition that there are uncountably many non-regular solutions. This poses an intriguing question: why a particular non-regular solution is always obtained in numerical simulations? This has important ramifications to the discussion of which intermediate shocks are really admissible.

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Shock-Capturing Approach and Nonevolutionary Solutions in Magnetohydrodynamics

Cited by 89 publications

References 9 publications

ECHO: a Eulerian conservative high-order scheme for general relativistic magnetohydrodynamics and magnetodynamics

ECHO: a Eulerian conservative high-order scheme for general relativistic magnetohydrodynamics and magnetodynamics

Quantitative Evaluation of Ion Kinetic Effect in Magnetic Field Inflation by the Injection of a Plasma Jet

Regular and non-regular solutions of the Riemann problem in ideal magnetohydrodynamics

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