Structural stability of shock waves and current-vortex sheets in shallow water magnetohydrodynamics

Trakhinin, Yuri

doi:10.1007/s00033-020-01344-2

Cited by 9 publications

(7 citation statements)

References 28 publications

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“…Therefore, the study of (2a)-(2f ) in mass Lagrangian coordinates is reduced to the analysis of a single equation (8).…”

Section: The One-dimensional Smhd Equations In Lagrangian Coordinatesmentioning

confidence: 99%

“…Notice that if a = 0, then (8) corresponds to the classical one-dimensional shallow water equation, considered in mass Lagrangian coordinates. Symmetries and conservation laws of this case (a = 0) have already been studied in [29] (see also [28,37]).…”

Section: The One-dimensional Smhd Equations In Lagrangian Coordinatesmentioning

confidence: 99%

“…The local well-posedness in time of the one-dimensional model was proven in [7]. The structural stability of shock waves and current-vortex sheets in the SMHD was studied in [8].…”

Section: Introductionmentioning

confidence: 99%

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Symmetries and conservation laws of the one-dimensional shallow water magnetohydrodynamics equations in Lagrangian coordinates

Капцов¹,

Meleshko²,

Дородницын³

2022

J. Phys. A: Math. Theor.

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Symmetries of the one-dimensional shallow water magnetohydrodynamics equations (SMHD) in Gilman's approximation are studied. The SMHD equations are considered in case of a plane and uneven bottom topography in Lagrangian and Eulerian coordinates. Symmetry classification separates out all bottom topographies which yields substantially different admitted symmetries. The SMHD equations in Lagrangian coordinates were reduced to a single second order PDE. The Lagrangian formalism and Noether's theorem are used to construct conservation laws of the SMHD equations. Some new conservation laws for various bottom topographies are obtained. The results are also represented in Eulerian coordinates. Invariant and partially invariant solutions are constructed.

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“…Therefore, the study of (2a)-(2f ) in mass Lagrangian coordinates is reduced to the analysis of a single equation (8).…”

Section: The One-dimensional Smhd Equations In Lagrangian Coordinatesmentioning

confidence: 99%

Section: The One-dimensional Smhd Equations In Lagrangian Coordinatesmentioning

confidence: 99%

See 1 more Smart Citation

Symmetries and conservation laws of the one-dimensional shallow water magnetohydrodynamics equations in Lagrangian coordinates

Капцов¹,

Meleshko²,

Дородницын³

2022

J. Phys. A: Math. Theor.

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“…In recent paper, Trakhinin [44] investigated the structural stability of shock waves and current-vortex sheets in the non-linear shallow MHD equation (4.1).…”

Section: Remarkmentioning

confidence: 99%

Asymptotic Shallow Models Arising in Magnetohydrodynamics

Alonso-Orán

2021

Water Waves

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In this paper, we derive new shallow asymptotic models for the free boundary plasma-vacuum problem governed by the magnetohydrodynamic equations which are vital when describing large-scale processes in flows of astrophysical plasma. More precisely, we present the magnetic analogue of the 2D Green–Naghdi equations for water waves under a weak magnetic pressure assumption in the presence of weakly sheared vorticity and magnetic currents. Our method is inspired by ideas for hydrodynamic flows developed in Castro and Lannes (2014) to reduce the three-dimensional dynamics of the vorticity and current to a finite cascade of two dimensional equations which can be closed at the precision of the model.

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“…The consideration of the shallow water equations in the presence of a magnetic field (SMHD) is a relatively new area of magnetohydrodynamics. One of the first models to describe SMHD was presented in [1] (the Gilman model), and since then, many papers by various authors have been devoted to studying this model, including its stability [2], numerical simulation [3][4][5][6][7], and conservation laws [8,9] (see, for example, a more detailed review in [8]).…”

Section: Introductionmentioning

confidence: 99%

Conservative invariant finite‐difference schemes for the modified shallow water equations in Lagrangian coordinates

2022

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The one-dimensional modified shallow water equations in Lagrangian coordinates are considered.Relations between symmetries and conservation laws in Lagrangian coordinates, in mass Lagrangian variables and Eulerian coordinates are discussed. For equations in Lagrangian coordinates, invariant finite-difference schemes are constructed. These schemes admit all conservation laws, which are related to Lie symmetries admitted by the differential model via Noether's theorem: they possess the difference analogs of the conservation laws of mass, momentum, energy, the law of center-of-mass motion for horizontal, inclined, and parabolic bottom topographies. The invariant conservative difference schemes are tested numerically in comparison with an ad hoc approximation-invariant scheme.

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Structural stability of shock waves and current-vortex sheets in shallow water magnetohydrodynamics

Cited by 9 publications

References 28 publications

Symmetries and conservation laws of the one-dimensional shallow water magnetohydrodynamics equations in Lagrangian coordinates

Symmetries and conservation laws of the one-dimensional shallow water magnetohydrodynamics equations in Lagrangian coordinates

Asymptotic Shallow Models Arising in Magnetohydrodynamics

Conservative invariant finite‐difference schemes for the modified shallow water equations in Lagrangian coordinates

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