Multi-dimensional hyperbolic partial differential equations

Benzoni-Gavage, Sylvie; Serre, Denis

doi:10.1093/acprof:oso/9780199211234.001.0001

Cited by 168 publications

(254 citation statements)

References 104 publications

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“…To analyze (1.7) one uses pseudo-differential calculus. We have the following well-posedness result, see [3], Theorem 2.6. …”

Section: The Continuous Problemmentioning

confidence: 98%

“…The standard procedure for hyperbolic equations in multi space dimensions is to symmetrize the system and derive an energy estimate. In order to do this we introduce the operator a · ∇ = a 1 ∂ x + a 2 ∂ y + a 3 Then we see that a simple way to symmetrize (1.1) is to add a "source" term: (which is supposed to be zero anyway!) −udivB, resulting in ∂ t B + curl(B × u) = −udivB.…”

Section: The Continuous Problemmentioning

confidence: 99%

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Stable upwind schemes for the magnetic induction equation

Fuchs¹,

Karlsen²,

Mishra³

et al. 2009

ESAIM: M2AN

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Abstract.We consider the magnetic induction equation for the evolution of a magnetic field in a plasma where the velocity is given. The aim is to design a numerical scheme which also handles the divergence constraint in a suitable manner. We design and analyze an upwind scheme based on the symmetrized version of the equations in the non-conservative form. The scheme is shown to converge to a weak solution of the equations. Furthermore, the discrete divergence produced by the scheme is shown to be bounded. We report several numerical experiments that show that the stable upwind scheme of this paper is robust.Mathematics Subject Classification. 65M12, 35L65.

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“…To analyze (1.7) one uses pseudo-differential calculus. We have the following well-posedness result, see [3], Theorem 2.6. …”

Section: The Continuous Problemmentioning

confidence: 98%

Section: The Continuous Problemmentioning

confidence: 99%

Stable upwind schemes for the magnetic induction equation

Fuchs¹,

Karlsen²,

Mishra³

et al. 2009

ESAIM: M2AN

View full text Add to dashboard Cite

show abstract

“…Note that standard approaches to hyperbolic PDEs such as the linearisation approach described in Benzoni et al [3], or the addition of non-degenerate artificial viscosity as described in Evans [14], are not directly applicable to our problem as they destroy the structure of the equations on which our energy estimate relies.…”

Section: Existing Results and Methodology For Similar Problemsmentioning

confidence: 99%

“…For a vector w ∈ R 3 , we will write w for the first 2 components of w, so that w = (w , w 3 ). For example, x = (x , x 3 ), so that x denotes horizontal position, u = (u , u 3 ), so that u denotes horizontal velocity, and ∇ = (∇ , ∂ x 3 ), so that ∇ denotes horizontal gradient.…”

Section: Notationmentioning

confidence: 99%

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Short-Time Structural Stability of Compressible Vortex Sheets with Surface Tension

Stevens

2016

Arch Rational Mech Anal

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Assume we start with an initial vortex-sheet configuration which consists of two inviscid fluids with density bounded below flowing smoothly past each other, where a strictly positive fixed coefficient of surface tension produces a surface tension force across the common interface, balanced by the pressure jump. We model the fluids by the compressible Euler equations in three space dimensions with a very general equation of state relating the pressure, entropy and density such that the sound speed is positive. We prove that, for a short time, there exists a unique solution of the equations with the same structure.The mathematical approach consists of introducing a carefully chosen artificial viscosity-type regularisation which allows one to linearise the system so as to obtain a collection of transport equations for the entropy, pressure and curl together with a parabolic-type equation for the velocity which becomes fairly standard after rotating the velocity according to the interface normal. We prove a high order energy estimate for the non-linear equations that is independent of the artificial viscosity parameter which allows us to send it to zero. This approach loosely follows that introduced by Shkoller et al. in the setting of a compressible liquid-vacuum interface.Although already considered by Coutand et al. [10] and Lindblad [17], we also make some brief comments on the case of a compressible liquid-vacuum interface, which is obtained from the vortex sheets problem by replacing one of the fluids by vacuum, where it is possible to obtain a structural stability result even without surface tension.

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Local existence for the free boundary problem for nonrelativistic and Relativistic compressible Euler equations with a vacuum boundary condition

Trakhinin

2009

Comm Pure Appl Math

238

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We study the free boundary problem for the equations of compressible Euler equations with a vacuum boundary condition. Our main goal is to recover in Eulerian coordinates the earlier well-posedness result obtained by Lindblad [11] for the isentropic Euler equations and extend it to the case of full gas dynamics.For technical simplicity we consider the case of an unbounded domain whose boundary has the form of a graph and make short comments about the case of a bounded domain. We prove the local-in-time existence in Sobolev spaces by the technique applied earlier to weakly stable shock waves and characteristic discontinuities [5,21]. It contains, in particular, the reduction to a fixed domain, using the "good unknown" of Alinhac [1], and a suitable Nash-Moser-type iteration scheme. A certain modification of such an approach is caused by the fact that the symbol associated to the free surface is not elliptic. This approach is still directly applicable to the relativistic version of our problem in the setting of special relativity and we briefly discuss its extension to general relativity.

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Multi-dimensional hyperbolic partial differential equations

Cited by 168 publications

References 104 publications

Stable upwind schemes for the magnetic induction equation

Stable upwind schemes for the magnetic induction equation

Short-Time Structural Stability of Compressible Vortex Sheets with Surface Tension

Local existence for the free boundary problem for nonrelativistic and Relativistic compressible Euler equations with a vacuum boundary condition

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