Exact Solutions of Stationary Equations of Ideal Magnetohydrodynamics in the Natural Coordinate System

Головин, С. В.; Sesma, L. Toledo

doi:10.1134/s0021894419020056

Cited by 7 publications

(5 citation statements)

References 11 publications

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“…The MHD equations are nonlinear, so that even in the one-dimensional case only their particular solutions are known [6][7][8][9][10]. Therefore, numerical modeling in magnetohydrodynamics is of great practical interest.…”

Section: Introductionmentioning

confidence: 99%

Invariant Finite-Difference Schemes for Plane One-Dimensional MHD Flows That Preserve Conservation Laws

Дородницын

Капцов

2022

Mathematics

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Invariant finite-difference schemes are considered for one-dimensional magnetohydrodynamics (MHD) equations in mass Lagrangian coordinates for the cases of finite and infinite conductivity. The construction of these schemes makes use of results of the group classification of MHD equations previously obtained by the authors. On the basis of the classical Samarskiy–Popov scheme, new schemes are constructed for the case of finite conductivity. These schemes admit all symmetries of the original differential model and have difference analogues of all of its local differential conservation laws. New, previously unknown, conservation laws are found using symmetries and direct calculations. In the case of infinite conductivity, conservative invariant schemes are constructed as well. For isentropic flows of a polytropic gas the proposed schemes possess the conservation law of energy and preserve entropy on two time layers. This is achieved by means of specially selected approximations for the equation of state of a polytropic gas. In addition, invariant difference schemes with additional conservation laws are proposed. A new scheme for the case of finite conductivity is tested numerically for various boundary conditions, which shows accurate preservation of difference conservation laws.

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Section: Introductionmentioning

confidence: 99%

Invariant Finite-Difference Schemes for Plane One-Dimensional MHD Flows That Preserve Conservation Laws

Дородницын

Капцов

2022

Mathematics

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“…The case of the infinite conductivity was examined in Ibragimov [8] and Paliathanasis [9]. Invariant solutions were considered in previous studies [10][11][12][13][14][15][16]. Comprehensive analysis of MGD equations in Eulerian and Lagrangian coordinates with plain and cylindrical symmetries were given in Dorodnitsyn et al [17,18].…”

Section: Introductionmentioning

confidence: 99%

Group classification of the two‐dimensional magnetogasdynamics equations in Lagrangian coordinates

Meleshko

Капцов

Moyo

et al. 2023

Math Methods in App Sciences

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The present paper is devoted to the group classification of magnetogasdynamics equations in which dependent variables in Euler coordinates depend on time and two spatial coordinates. It is assumed that the continuum is inviscid and nonthermal polytropic gas with infinite electrical conductivity. The equations are considered in mass Lagrangian coordinates. Use of Lagrangian coordinates allows reducing number of dependent variables. The analysis presented in this article gives complete group classification of the studied equations. This analysis is necessary for constructing invariant solutions and conservation laws on the base of Noether's theorem.

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“…The case of the infinite conductivity was examined in [8,9]. Invariant solutions were considered in [10][11][12][13][14][15][16]. Comprehensive analysis of MGD equations in Eulerian and Lagrangian coordinates with plain and cylindrical symmetries were given in [17,18].…”

Section: Introductionmentioning

confidence: 99%

Group classification of the two-dimensional magnetogasdynamics equations in Lagrangian coordinates

Meleshko

Kaptsov

Moyo

et al. 2022

Preprint

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Exact Solutions of Stationary Equations of Ideal Magnetohydrodynamics in the Natural Coordinate System

Cited by 7 publications

References 11 publications

Invariant Finite-Difference Schemes for Plane One-Dimensional MHD Flows That Preserve Conservation Laws

Invariant Finite-Difference Schemes for Plane One-Dimensional MHD Flows That Preserve Conservation Laws

Group classification of the two‐dimensional magnetogasdynamics equations in Lagrangian coordinates

Group classification of the two-dimensional magnetogasdynamics equations in Lagrangian coordinates

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