Multi-symplectic, Lagrangian, one-dimensional gas dynamics

Webb, G. M.

doi:10.1063/1.4919669

Cited by 7 publications

(22 citation statements)

References 51 publications

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“…Proof.The proof is essentially the same as that given by Webb (2015) for the case of 1D gas dynamics. A critical component of the proof is the use of Cartan's magic formula:…”

Section: Extensions Comments and Correctionsmentioning

confidence: 73%

“…The fourth term on the right handside of (4.13) was missed in equation (5.48) in Webb et al (2014c). Also in (4.13) we used the identity: A consistent approach to the multisymplectic equations using differential forms for 1D Lagrangian gas dynamics was given by Webb (2015). Webb and Anco (2015) have given the corresponding theory for multidimensional, ideal, compressible, Lagrangian gas dynamics.…”

Section: Extensions Comments and Correctionsmentioning

confidence: 99%

“…The ideal is closed if dβ i = c ij ∧ β z j , where the c ij are forms. It should be noted that the ideal of forms obtained by Webb (2015) for 1-D, Lagrangian, multi-symplectic gas dynamics is closed. The ideal of forms for multi-dimensional, Lagrangian, compressible gas dynamics obtained from the Cartan-Poincaré form is also a closed ideal (Webb & Anco 2015).…”

Section: Remark the Integrability Conditions D Dω αmentioning

confidence: 99%

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Multi-symplectic magnetohydrodynamics: II, addendum and erratum

2015

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A recent paper by Webb et al. (2014c) on multi-symplectic magnetohydrodynamics (MHD) using Clebsch variables in an Eulerian action principle with constraints is further extended. We relate a class of symplecticity conservation laws to a vorticity conservation law, and provide a corrected form of the Poincaré-Cartan differential form formulation of the system. We also correct some typographical errors (omissions) in Webb et al. (2014c). We show that the vorticity-symplecticity conservation law, that arises as a compatibility condition on the system, expressed in terms of the Clebsch variables is equivalent to taking the curl of the conservation form of the MHD momentum equation. We use the Cartan-Poincaré form to obtain a class of differential forms that represent the system using Cartan's geometric theory of partial differential equations.

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“…Proof.The proof is essentially the same as that given by Webb (2015) for the case of 1D gas dynamics. A critical component of the proof is the use of Cartan's magic formula:…”

Section: Extensions Comments and Correctionsmentioning

confidence: 73%

Section: Extensions Comments and Correctionsmentioning

confidence: 99%

Section: Remark the Integrability Conditions D Dω αmentioning

confidence: 99%

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Multi-symplectic magnetohydrodynamics: II, addendum and erratum

2015

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“…as an equivalent form of ℓ m where w(p, S) = w(ρ, S) is the enthalpy of the gas. Note that the Lagrangian lm in (3.33) has the functional form: (3.34) and that wp = τ and wS = T (3.35) (see Webb (2015)).…”

Section: The Euler Lagrange Equationsmentioning

confidence: 99%

“…The de Donder Weyl Hamiltonian equations apply to action principles in which the Lagrangian L = L(x, ϕ i , ∂ϕ i /∂x µ ) where x are the independent variables and the ϕ k are the dependent variables (1 ≤ k ≤ m say, k integer), which includes at least two independent partial derivatives ∂ϕ k /∂x s , (1 ≤ s ≤ n, n ≥ 2). Webb (2015) cast the equations of ideal, 1D, Lagrangian gas dynamics in the de Donder-Weyl Hamiltonian form. In this development, the dependent variables are the Eulerian particle position x = x(m, t) and gas entropy S = S(m, t) where S t = 0.…”

Section: Introductionmentioning

confidence: 99%

Vorticity and Symplecticity in Multi-Symplectic Lagrangian Gas Dynamics

Webb,

Anco

2016

Preprint

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The Lagrangian, multi-dimensional, ideal, compressible gasdynamic equations are written in a multi-symplectic form, in which the Lagrangian fluid labels, m i (the Lagrangian mass coordinates) and time t are the independent variables, and in which the Eulerian position of the fluid element x = x(m, t) and the entropy S = S(m, t) are the dependent variables. Constraints in the variational principle are incorporated by means of Lagrange multipliers. The constraints are: the entropy advection equation S t = 0, the Lagrangian map equation x t = u where u is the fluid velocity, and the mass continuity equation which has the form J = τ where J = det(x ij ) is the Jacobian of the Lagrangian map in which x ij = ∂x i /∂m j and τ = 1/ρ is the specific volume of the gas. The internal energy per unit volume of the gas ε = ε(ρ, S) corresponds to a non-barotropic gas. The Lagrangian is used to define multi-momenta, and to develop de-Donder Weyl Hamiltonian equations. The de Donder Weyl equations are cast in a multi-symplectic form. The pullback conservation laws and the symplecticity conservation laws are obtained. One class of symplecticity conservation laws give rise to vorticity and potential vorticity type conservation laws, and another class of symplecticity laws are related to derivatives of the Lagrangian energy conservation law with respect to the Lagrangian mass coordinates m i . We show that the vorticitysymplecticity laws can be derived by a Lie dragging method, and also by using Noether's second theorem and a fluid relabelling symmetry which is a divergence symmetry of the action. We obtain the Cartan-Poincaré form describing the equations and we discuss a set of differential forms representing the equation system.

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Generalized Riemann waves and their adjoinment through a shock wave

Chaiyasena

Worapitpong

Meleshko

2018

Math. Model. Nat. Phenom.

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Generalized simple waves of the gas dynamics equations in Lagrangian and Eulerian descriptions are studied in the paper. As in the collision of a shock wave and a rarefaction wave, a flow becomes nonisentropic. Generalized simple waves are applied to describe such flows. The first part of the paper deals with constructing a solution describing their adjoinment through a shock wave in Eulerian coordinates. Even though the Eulerian form of the gas dynamics equations is most frequently used in applications, there are advantages for some problems concerning the gas dynamics equations in Lagrangian coordinates, for example, of being able to be reduced to an Euler–Lagrange equation. Through the technique of differential constraints, necessary and sufficient conditions for the existence of generalized simple waves in the Lagrangian description are provided in the second part of the paper.

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Multi-symplectic, Lagrangian, one-dimensional gas dynamics

Cited by 7 publications

References 51 publications

Multi-symplectic magnetohydrodynamics: II, addendum and erratum

Multi-symplectic magnetohydrodynamics: II, addendum and erratum

Vorticity and Symplecticity in Multi-Symplectic Lagrangian Gas Dynamics

Generalized Riemann waves and their adjoinment through a shock wave

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