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Kinetic functions in magnetohydrodynamics with resistivity and hall effects

by Ph. LeFloch and S. Mishra

(Report number 2009-36)

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 here the associated kinetic function which characterizes the dynamics of undercompressive shocks driven by resistivity and Hall effects. To this end, we design here a new class of schemes with controled dissipation, following recent work by LeFloch and Mohammadian. The equivalent equation associated with a scheme provides a guideline to able to capture physically relevant shocks. We propose a class schemes based on highorder entropy conservative, finite differences for the hyperbolic flux and on high-order central differences for the resistivity and Hall terms. The schemes are tested for several regimes of initial data (co-planar or not) and parameter values, and allow us to analyze the properties of nonclassical shocks and establish the existence of monotone kinetic function in magnetohydrodynamics.

Keywords:

@Techreport{LM09_417,
  author = {Ph. LeFloch and S. Mishra},
  title = {Kinetic functions in magnetohydrodynamics with resistivity and hall effects},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2009-36},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2009/2009-36.pdf },
  year = {2009}
}

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