Stable upwind schemes for the magnetic induction equation

Franz G. Fuchs; Kenneth H. Karlsen; Siddharta Mishra; Nils H. Risebro

doi:10.1051/m2an/2009006

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Volume 43 / No 5 (September-October 2009)

ESAIM: M2AN, 43 5 (2009) 825-852

References

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Issue		ESAIM: M2AN Volume 43, Number 5, September-October 2009


Page(s)		825 - 852
DOI		https://doi.org/10.1051/m2an/2009006
Published online		08 April 2009

D.S. Balsara and D. Spicer, A staggered mesh algorithm using high order Godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamic simulations. J. Comp. Phys. 149 (1999) 270–292. [NASA ADS] [CrossRef] [MathSciNet]
T.J. Barth, Numerical methods for gas dynamics systems, in An introduction to recent developments in theory and numerics for conservation laws, D. Kröner, M. Ohlberger and C. Rohde Eds., Springer (1999).
S. Benzoni-Gavage and D. Serre, Multidimensional hyperbolic, Partial differential equations – First-order systems and applications, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford (2007).
N. Besse and D. Kröner, Convergence of the locally divergence free discontinuous Galerkin methods for induction equations for the 2D-MHD system. ESAIM: M2AN 39 (2005) 1177–1202. [CrossRef] [EDP Sciences]
J.U. Brackbill and D.C. Barnes, The effect of nonzero divB on the numerical solution of the magnetohydrodynamic equations. J. Comp. Phys. 35 (1980) 426–430. [NASA ADS] [CrossRef] [MathSciNet]
W. Dai and P.R. Woodward, A simple finite difference scheme for multi-dimensional magnetohydrodynamic equations. J. Comp. Phys. 142 (1998) 331–369. [NASA ADS] [CrossRef]
C. Evans and J.F. Hawley, Simulation of magnetohydrodynamic flow: a constrained transport method. Astrophys. J. 332 (1998) 659. [NASA ADS] [CrossRef]
F.G. Fuchs, S. Mishra and N.H. Risebro, Splitting based finite volume schemes for ideal MHD equations. J. Comp. Phys. 228 (2009) 641–660. [CrossRef]
S.K. Godunov, The symmetric form of magnetohydrodynamics equation. Num. Meth. Mech. Cont. Media 1 (1972) 26–34.
J.D. Jackson, Classical Electrodynamics. Third Edn., Wiley (1999).
V. Jovanovič and C. Rohde, Finite volume schemes for Friedrichs systems in multiple space dimensions: a priori and a posteriori error estimates. Num. Meth. PDE 21 (2005) 104–131.
R.J. LeVeque, Finite volume methods for hyperbolic problems. Cambridge University Press (2002).
G.K. Parks, Physics of Space Plasmas: An Introduction. Addition-Wesley (1991).
K.G. Powell, A approximate Riemann solver for magnetohydrodynamics (that works in more than one dimension). Technical report 94-24, ICASE, Langley, VA, USA (1994).
K.G. Powell, P.L. Roe, T.J. Linde, T.I. Gombosi and D.L. De Zeeuw, A solution adaptive upwind scheme for ideal MHD. J. Comp. Phys. 154 (1999) 284–309. [NASA ADS] [CrossRef]
J. Rossmanith, A wave propagation method with constrained transport for shallow water and ideal magnetohydrodynamics. Ph.D. Thesis, University of Washington, Seattle, USA (2002).
D.S. Ryu, F. Miniati, T.W. Jones and A. Frank, A divergence free upwind code for multidimensional magnetohydrodynamic flows. Astrophys. J. 509 (1998) 244–255. [NASA ADS] [CrossRef]
M. Torrilhon, Locally divergence preserving upwind finite volume schemes for magnetohyrodynamic equations. SIAM. J. Sci. Comp. 26 (2005) 1166–1191. [CrossRef]
M. Torrilhon and M. Fey, Constraint-preserving upwind methods for multidimensional advection equations. SIAM. J. Num. Anal. 42 (2004) 1694–1728. [CrossRef] [MathSciNet]
G. Toth, The divB = 0 constraint in shock capturing magnetohydrodynamics codes. J. Comp. Phys. 161 (2000) 605–652. [NASA ADS] [CrossRef] [MathSciNet]
J-P. Vila and P. Villedeau, Convergence of explicit finite volume scheme for first order symmetric systems. Numer. Math. 94 (2003) 573–602. [CrossRef] [MathSciNet]

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[1] D.S. Balsara and D. Spicer, A staggered mesh algorithm using high order Godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamic simulations. J. Comp. Phys. 149 (1999) 270–292. [NASA ADS] [CrossRef] [MathSciNet]

[2] T.J. Barth, Numerical methods for gas dynamics systems, in An introduction to recent developments in theory and numerics for conservation laws, D. Kröner, M. Ohlberger and C. Rohde Eds., Springer (1999).

[3] S. Benzoni-Gavage and D. Serre, Multidimensional hyperbolic, Partial differential equations – First-order systems and applications, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford (2007).

[4] N. Besse and D. Kröner, Convergence of the locally divergence free discontinuous Galerkin methods for induction equations for the 2D-MHD system. ESAIM: M2AN 39 (2005) 1177–1202. [CrossRef] [EDP Sciences]

[5] J.U. Brackbill and D.C. Barnes, The effect of nonzero divB on the numerical solution of the magnetohydrodynamic equations. J. Comp. Phys. 35 (1980) 426–430. [NASA ADS] [CrossRef] [MathSciNet]

[6] W. Dai and P.R. Woodward, A simple finite difference scheme for multi-dimensional magnetohydrodynamic equations. J. Comp. Phys. 142 (1998) 331–369. [NASA ADS] [CrossRef]

[7] C. Evans and J.F. Hawley, Simulation of magnetohydrodynamic flow: a constrained transport method. Astrophys. J. 332 (1998) 659. [NASA ADS] [CrossRef]

[8] F.G. Fuchs, S. Mishra and N.H. Risebro, Splitting based finite volume schemes for ideal MHD equations. J. Comp. Phys. 228 (2009) 641–660. [CrossRef]

[9] S.K. Godunov, The symmetric form of magnetohydrodynamics equation. Num. Meth. Mech. Cont. Media 1 (1972) 26–34.

[10] J.D. Jackson, Classical Electrodynamics. Third Edn., Wiley (1999).

[11] V. Jovanovič and C. Rohde, Finite volume schemes for Friedrichs systems in multiple space dimensions: a priori and a posteriori error estimates. Num. Meth. PDE 21 (2005) 104–131.

[12] R.J. LeVeque, Finite volume methods for hyperbolic problems. Cambridge University Press (2002).

[13] G.K. Parks, Physics of Space Plasmas: An Introduction. Addition-Wesley (1991).

[14] K.G. Powell, A approximate Riemann solver for magnetohydrodynamics (that works in more than one dimension). Technical report 94-24, ICASE, Langley, VA, USA (1994).

[15] K.G. Powell, P.L. Roe, T.J. Linde, T.I. Gombosi and D.L. De Zeeuw, A solution adaptive upwind scheme for ideal MHD. J. Comp. Phys. 154 (1999) 284–309. [NASA ADS] [CrossRef]

[16] J. Rossmanith, A wave propagation method with constrained transport for shallow water and ideal magnetohydrodynamics. Ph.D. Thesis, University of Washington, Seattle, USA (2002).

[17] D.S. Ryu, F. Miniati, T.W. Jones and A. Frank, A divergence free upwind code for multidimensional magnetohydrodynamic flows. Astrophys. J. 509 (1998) 244–255. [NASA ADS] [CrossRef]

[18] M. Torrilhon, Locally divergence preserving upwind finite volume schemes for magnetohyrodynamic equations. SIAM. J. Sci. Comp. 26 (2005) 1166–1191. [CrossRef]

[19] M. Torrilhon and M. Fey, Constraint-preserving upwind methods for multidimensional advection equations. SIAM. J. Num. Anal. 42 (2004) 1694–1728. [CrossRef] [MathSciNet]

[20] G. Toth, The divB = 0 constraint in shock capturing magnetohydrodynamics codes. J. Comp. Phys. 161 (2000) 605–652. [NASA ADS] [CrossRef] [MathSciNet]

[21] J-P. Vila and P. Villedeau, Convergence of explicit finite volume scheme for first order symmetric systems. Numer. Math. 94 (2003) 573–602. [CrossRef] [MathSciNet]