Acta mathematica scientia,Series B ›› 2009, Vol. 29 ›› Issue (6): 1684-1702.doi: 10.1016/S0252-9602(10)60011-2

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KINETIC FUNCTIONS IN MAGNETOHYDRODYNAMICS WITH RESISTIVITY AND HALL EFFECT

 Philippe G. LeFloch, Siddhartha Mishra   

  1. Laboratoire Jacques-Louis Lions, |Centre National de la Recherche Scientifique, Universite Pierre et Marie Curie (Paris 6), 4 Place Jussieu, 75258 Paris, France;Seminar for Applied Mathematics, D-Math, ETH, HG, Raemistrasse, Zurich-8092, Switzerland
  • Received:2009-10-31 Online:2009-11-20 Published:2009-11-20
  • Supported by:

    The first author (PLF) was partially supported by the Centre National de la Recherche Scientifique (CNRS)and the Agence Nationale de la Recherche (ANR).

Abstract:

We consider a nonlinear hyperbolic system of two conservation laws which arises in ideal magnetohydrodynamics and includes second-order
terms accounting for magnetic resistivity and Hall effect. We show that the initial value problem for this model may lead to solutions exhibiting
complex wave structures, including undercompressive nonclassical shock waves. We investigate numerically the subtle competition that takes place between the hyperbolic, diffusive, and dispersive parts of the system. Following Abeyratne, Knowles, LeFloch, and Truskinovsky, who studied similar questions arising in fluid and solid flows, we determine the associated kinetic function which characterizes the dynamics of undercompressive shocks driven by resistivity and Hall effect. To this end, we design a new class of ``schemes with controled dissipation'',
following recent work by LeFloch and Mohammadian. It is now recognized that the equivalent equation associated with a scheme provides a guideline to design schemes that capture physically relevant, nonclassical shocks. We propose a new class of schemes based on high-order entropy conservative, finite differences for the hyperbolic flux, and  high-order central differences for the resistivity and Hall terms. These schemes are tested for several regimes of (co-planar or not) initial data and parameter values, and allow us to analyze the properties of nonclassical shocks and establish the existence of monotone kinetic functions in magnetohydrodynamics.

Key words: hyperbolic conservation law, magnetohydrodynamics, magnetic resistivity, Hall effect, undercompressive shock, kinetic function

CLC Number: 

  • 35L65
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