Acta mathematica scientia, Series B >
KINETIC FUNCTIONS IN MAGNETOHYDRODYNAMICS WITH RESISTIVITY AND HALL EFFECT
Received date: 2009-10-31
Online 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).
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.
Philippe G. LeFloch , Siddhartha Mishra . KINETIC FUNCTIONS IN MAGNETOHYDRODYNAMICS WITH RESISTIVITY AND HALL EFFECT[J]. Acta mathematica scientia, Series B, 2009 , 29(6) : 1684 -1702 . DOI: 10.1016/S0252-9602(10)60011-2
[1] Abeyratne R, Knowles J K. Kinetic relations and the propagation of phase boundaries in solids. Arch Rational Mech Anal, 1991, 114: 119--154
[2] Abeyratne R, Knowles J K. Implications of viscosity and strain-gradient effects for the kinetics of propagating phase boundaries in solids.
SIAM J Appl Math, 1991, 51: 1205--1221
[3] Avezedo A, Marchesin D, Plohr B J, Zumbrun K. Non-uniqueness of solutions of Riemann problems caused by 2-cycles of shock saves//Proc of Fifth International Conf on Hyperbolic Equations. Singapore: World Scientific, 1996: 43--51
[4] Bertozzi A, Munch A, Shearer M. Undercompressive shocks in thin film flows. Phys D, 1999, 134: 431--464
[5] Bertozzi A, Shearer M. Existence of undercompressive traveling waves in thin film equations. SIAM J Math Anal, 2000, 32: 194--213
[6] Biskamp D. Nonlinear magnetohydrodynamics, Cambridge monographs on plasma physics. Cambridge Univ Press, 1993
[7] Brio M, Hunter J. Rotationally invariant hyperbolic waves. Comm Pure Appl Math, 1990, 43: 1037--1053
[8] Chalons C, LeFloch P G. A fully discrete scheme for diffusive-dispersive conservation laws. Numerische Math, 2001, 89: 493--509
[9] Chalons C, LeFloch P G. High-order entropy conservative schemes and kinetic relations for van der Waals fluids. J Comput Phys, 2001, 167: 1--23
[10] Cohen R H, Kulsrud R M. Non-linear evolution of quasi-parallel propagating hydro-magnetic waves. Phys Fluid, 1974, 17: 2215--2225
[11] Fan H T, Slemrod M. The Riemann problem for systems of conservation laws of mixed type//Shock Induced Transitions and Phase Structures in General Media. IMA Vol Math Appl 52. New York: Springer-Verlag, 1993: 61--91
[12] Fjordholm U S, Mishra S, Tadmor E. Energy preserving and energy stable schemes for the shallow water equations//Cucker F, Pinkus A, Todd M, eds. Foundations of Computational Mathematics, Hong Kong 2008. London Math Soc Lecture Notes Ser 363. London Math Soc, 2009: 93--139
[13] Freist\"uhler H. Dynamical stability and vanishing viscosity: a case study of a non-strictly hyperbolic system of conservation laws. Comm Pure Appl Math, 1992, 45: 561--582
[14] Freist\"uhler H. Stability of nonclassical shock waves//Proc of Fifth International Conference on Hyperbolic Equations. Singapore: World Scientific, 1996: 120--129
[15] Freistuhler H, Pitman E B. A numerical study of a rotationally degenerate hyperbolic system, Part 1: The Riemann problem. J Comput Phys, 1992, 100: 306--321
[16] Hayes B T, LeFloch P G. Nonclassical shocks and kinetic relations: Scalar conservation laws. Arch Rational Mech Anal, 1997, 139: 1--56
[17] Hayes B T, LeFloch P G. Nonclassical shocks and kinetic relations: Finite difference schemes. SIAM J Num Anal, 1998, 35: 2169--2194
[18] Hayes B T, LeFloch P G. Nonclassical shocks and kinetic relations: Strictly hyperbolic systems. SIAM J Math Anal, 2000, 31: 941--991
[19] Jacobs D, McKinney W R, Shearer M. Traveling wave solutions of a modified Korteweg De-Vries equation. J Diff Eqns, 1995, 116: 448--467
[20] Keyfitz B, Kranzer H. A system of nonstrictly hyperbolic conservation laws arising in elasticity theory. Arch Rational Mech Anal, 1980, 72: 219--241
[21] Lax P D. Hyperbolic systems of conservation laws, II. Comm Pure Appl Math, 1957, 10: 537--566
[22] Lax P D. Hyperbolic systems of conservation laws and the mathematical theory of shock waves//Regional Confer Series in Appl Math 11. Philadelphia: SIAM, 1973
[23] LeFloch P G. Propagating phase boundaries: formulation of the problem and existence via the Glimm scheme. Arch Rational Mech Anal, 1993, 123: 153--197
[24] LeFloch P G. An introduction to nonclassical shocks of systems of conservation laws//International School on Hyperbolic Problems, Freiburg, Germany, Oct 97. Kroner D, Ohlberger M, Rohde C, eds. Lect Notes Comput Eng, Vol 5. Springer-Verlag, 1999: 28--72
[25] LeFloch P G. Hyperbolic Systems of Conservation Laws: The Theory of Classical and Nonclassical Shock Waves. Lectures in Mathematics, ETH Zurich. Birkh\"auser, 2002
[26] LeFloch P G. Kinetic relations for undercompressive shock waves. Physical, mathematical, and numerical issues. Centre for Advanced Study of the Norwegian Academy of Science and Letters. Holden H, Karlsen K, ed. 2010
[27] LeFloch P G, Mercier J M, Rohde C. Fully discrete entropy conservative schemes of arbitrary order. SIAM J Numer Anal, 2002, 40: 1968--1992
[28] LeFloch P G, Mohammadian M. Why many theories of shock waves are necessary. Kinetic functions, equivalent equations and fourth-order models. J Comput Phys, 2008, 27: 4162--4189
[29] LeFloch P G, Rohde C. High order schemes, entropy inequalities and nonclassical shocks. SIAM J Numer Anal, 2000, 37: 2023--2060
[30] LeFloch P G, Shearer M. Nonclassical Riemann solvers with nucleation. Proc Royal Soc Edinburgh, 2004, 134A：941-964
[31] Levy R, Shearer M. Comparison of two dynamic contact line models for driven thin liquid films. European J Appl Math, 2004, 15: 625--642
[32] Liu T P. The Riemann problem for general 2×2 conservation laws. Trans Amer Math Soc, 1974, 199: 89--112
[33] LeVeque R J. Finite Volume Methods for Hyperbolic Problems. Cambridge: Cambridge Univ Press, 2002
[34] Myong R S, Roe P L. Shock waves and rarefaction waves in magnetohydrodynamics, Part 1. A model system. J Plasma Phys, 1997, 58: 485--519
[35] Roe P L, Balsara D S. Note on the eigensystem of magnetohydrodynamics. SIAM J Appl Math, 1996, 56: 57--67
[36] Slemrod M. Admissibility criteria for propagating phase boundaries in a van der Waals fluid. Arch Rational Mech Anal, 1983, 81: 301--315
[37] Slemrod M. A limiting viscosity approach to the Riemann problem for materials exhibiting change of phase. Arch Rational Mech Anal, 1989, 105: 327--365
[38] Tadmor E. The numerical viscosity of entropy stable schemes for systems of conservation laws, I. Math Comp, 1987, 49: 91--103
[39] Tadmor E. Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems.
Acta Numerica, 2003, 12: 451--512
[40] Torrilhon M. Uniqueness conditions for Riemann problems of ideal magneto-hydrodynamics. J Plasma Phys, 2003, 69: 253--276
[41] Torrilhon M. Non-uniform convergence of finite volume schemes for Riemann problems of ideal magnetohydrodynamics. J Comput Phys, 2003, 192: 73--94
[42] Truskinovsky J. Dynamics of non-equilibrium phase boundaries in a heat conducting nonlinear elastic medium. J Appl Math and Mech (PMM), 1987, 51: 777--784
[43] Truskinovsky L. Kinks versus shocks//Fosdick R, Dunn E, Slemrod M, ed. Shock Induced Transitions and Phase Structures in General Media. IMA Vol Math Appl, 52. New York: Springer-Verlag, 1993: 185--229
[44] Wu C C, Kennel C F. The small amplitude magnetohydrodynamic Riemann problem. Phys Fluids B, 1993, 5: 2877--2886
/
| 〈 |
|
〉 |