2012, 5(2): 345-367. doi: 10.3934/dcdss.2012.5.345

High-order dimensionally split Lagrange-remap schemes for ideal magnetohydrodynamics

1. 

CEA, DAM, DIF, F-91297 Arpajon, France, France, France

2. 

INRIA Nancy-Grand Est, 615 rue du Jardin Botanique, 54600 Villers-lès-Nancy, IRMA, Université de Strasbourg, 7 rue René-Descartes, 67084 Strasbourg Cedex

Received  July 2009 Revised  February 2010 Published  September 2011

We first propose a new class of high-order finite volume schemes for solving the 1-D ideal magnetohydrodynamics equations that is particularly well-suited for modern computer architectures. Applicable to arbitrary equations of state, these schemes, based on a Lagrange-remap approach, are high-order accurate in both space and time in the non-linear regime. A multidimensional extension on 2-D Cartesian grids using a high-order dimensional splitting technique is then proposed. Numerical results up to fourth-order on smooth and non-smooth test problems are also provided.
Citation: Marc Wolff, Stéphane Jaouen, Hervé Jourdren, Eric Sonnendrücker. High-order dimensionally split Lagrange-remap schemes for ideal magnetohydrodynamics. Discrete & Continuous Dynamical Systems - S, 2012, 5 (2) : 345-367. doi: 10.3934/dcdss.2012.5.345
References:
[1]

D. S. Balsara, Second order accurate schemes for magnetohydrodynamics with divergence-free reconstruction,, Astrophys. J. Suppl. Ser., 151 (2004), 149. doi: 10.1086/381377.

[2]

W. Dai and P. R. Woodward, An approximate Riemann solver for ideal magnetohydrodynamics,, J. Comp. Phys., 111 (1994), 354. doi: 10.1006/jcph.1994.1069.

[3]

D. Ryu and T. W. Jones, Numerical magnetohydrodynamics in astrophysics: Algorithm and tests for one-dimensional flow,, The Astrophys. J., 442 (1995), 228. doi: 10.1086/175437.

[4]

S. A. E. G. Falle, S. S. Komissarov and P. Joarder, A multidimensional upwind scheme for magnetohydrodynamics,, Monthly Notices of the Royal Astronomical Society, 297 (1998), 265. doi: 10.1046/j.1365-8711.1998.01506.x.

[5]

R. K. Crockett, P. Colella, R. T. Fisher, R. J. Klein and C. I. McKee, An unsplit cell-centered Godunov method for ideal MHD,, J. Comp. Phys., 203 (2005), 422. doi: 10.1016/j.jcp.2004.08.021.

[6]

D. S. Balsara, Divergence-free reconstruction of magnetic fields and WENO schemes for magnetohydrodynamics,, J. Comp. Phys., 228 (2008), 5040. doi: 10.1016/j.jcp.2009.03.038.

[7]

A. Zachary, A. Malagoli and P. Colella, A higher-order Godunov method for multidimensional ideal magnetohydrodynamics,, SIAM J. Sci. Comp., 15 (1994), 263. doi: 10.1137/0915019.

[8]

G.-S. Jiang and C.-C. Wu, A high-order WENO finite difference scheme for the equation of ideal magnetohydrodynamics,, J. Comp. Phys., 150 (1999), 561. doi: 10.1006/jcph.1999.6207.

[9]

D. S. Balsara, T. Rumpf, M. Dumbser and C.-D. Munz, Efficient, high-accuracy ADER-WENO schemes for hydrodynamics and divergence-free magnetohydrodynamics,, J. Comp. Phys., 228 (2009), 2480. doi: 10.1016/j.jcp.2008.12.003.

[10]

F. Duboc, C. Enaux, S. Jaouen, H. Jourdren and M. Wolff, High-order dimensionally split Lagrange-remap schemes for compressible hydrodynamics,, C. R. Acad. Sci. Paris, 348 (2010), 105. doi: 10.1016/j.crma.2009.12.008.

[11]

T. D. Arber, A. W. Longbottom, C. L. Gerrard and A. M. Milne, A staggered grid, Lagrangian-Eulerian remap code for 3-D MHD simulations,, J. Comp. Phys., 171 (2001), 151. doi: 10.1006/jcph.2001.6780.

[12]

S. Del Pino and H. Jourdren, Arbitrary high-order schemes for the linear advection and wave equations: Application to hydrodynamics and aeroacoustics,, C. R. Math. Acad. Sci. Paris, 342 (2006), 441. doi: 10.1016/j.crma.2006.01.013.

[13]

P. Colella, Multidimensional upwind methods for hyperbolic conservation laws,, J. Comp. Phys., 87 (1990), 171. doi: 10.1016/0021-9991(90)90233-Q.

[14]

J. U. Brackbill and D. C. Barnes, The effect of nonzero $\nabla \cdot B$ on the numerical solution of the magnetohydrodynamics equations,, J. Comp. Phys., 35 (1980), 426. doi: 10.1016/0021-9991(80)90079-0.

[15]

S. H. Brecht, J. G. Lyon, J. A. Fedder and K. Hain, A simulation study of east-west IMF effects on the magnetosphere,, Geophysical Research Letter, 8 (1981), 397. doi: 10.1029/GL008i004p00397.

[16]

C. R. DeVore, Flux-corrected transport techniques for multidimensional compressible magnetohydrodynamics,, J. Comp. Phys., 92 (1991), 142. doi: 10.1016/0021-9991(91)90295-V.

[17]

W. Dai and P. R. Woodward, On the divergence-free condition and conservation laws in numerical simulations for supersonic magnetohydrodynamic flows,, Astrophys. J., 494 (1998), 317. doi: 10.1086/305176.

[18]

D. Ryu, F. Miniati, T. W. Jones and A. Frank, A divergence-free upwind code for multidimensional magnetohydrodynamic flows,, The Astrophys. J., 509 (1998), 244. doi: 10.1086/306481.

[19]

D. S. Balsara and D. S. Spicer, Maintaining pressure positivity in magnetohydrodynamic simulations,, J. Comp. Phys., 148 (1999), 133. doi: 10.1006/jcph.1998.6108.

[20]

D. S. Balsara and D. S. Spicer, A staggered mesh algorithm using high order Godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamic simulations,, J. Comp. Phys., 149 (1999), 270. doi: 10.1006/jcph.1998.6153.

[21]

P. Londrillo and L. DelZanna, On the divergence-free condition in Godunov-type schemes for ideal magnetohydrodynamics: The upwind constrained transport method,, J. Comp. Phys., 195 (2004), 17. doi: 10.1016/j.jcp.2003.09.016.

[22]

M. Torrilhon, Locally divergence-preserving upwind finite volume schemes for magnetohydrodynamic equations,, SIAM J. Sci. Comp., 26 (2005), 1166. doi: 10.1137/S1064827503426401.

[23]

K. G. Powell, An approximate Riemann solver for MHD (that works in more than one dimension),, ICASE Report 94-24, (1994), 94.

[24]

A. Dedner, F. Kemm, D. Kröner, C.-D. Munz, T. Schnitzer and M. Wesenberg, Hyperbolic divergence cleaning for MHD equations,, J. Comp. Phys., 175 (2002), 645. doi: 10.1006/jcph.2001.6961.

[25]

G. Tóth, The $\nabla \cdot B=0$ constraint in shock-capturing magnetohydrodynamics codes,, J. Comp. Phys., 161 (2000), 605. doi: 10.1006/jcph.2000.6519.

[26]

C.-W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws,, In, 1697 (1997), 325.

[27]

E. Forest and R. D. Ruth, Fourth-order symplectic integration,, Physica D, 43 (1990), 105.

[28]

R. I. McLachlan and P. Atela, The accuracy of symplectic integrators,, Nonlinearity, 5 (1992), 541. doi: 10.1088/0951-7715/5/2/011.

[29]

S. A. Chin, Forward and non-forward symplectic integrators in solving classical dynamics problems,, Internat. J. of Comp. Math., 84 (2007), 729. doi: 10.1080/00207160701458476.

[30]

A. Cook, Artificial fluid properties for large-eddy simulation of compressible turbulent mixing,, Phys. of Fluids, 19 (2007).

[31]

N. E. L. Haugen, Hydrodynamic and hydromagnetic energy spectra from large eddy simulations,, Phys. of Fluids, 18 (2006).

[32]

M. Germano, U. Piomelli, P. Moin and W. Cabot, A dynamic subgrid-scale eddy-viscosity model,, Phys. of Fluids, 3 (1991), 1760. doi: 10.1063/1.857955.

[33]

P. Picard, Reduction and exact solutions of the ideal magnetohydrodynamics equations,, Mathemat. Phys. e-prints, (2005).

[34]

A. Orszag and C. M. Tang, Small-scale structure of two-dimensional magnetohydrodynamic turbulence,, J. Fluid. Mech., 90 (1979), 129. doi: 10.1017/S002211207900210X.

[35]

R. B. Dahlburg and J. M. Picone, Evolution of the Orszag-Tang vortex system in a compressible medium. I. Initial average subsonic flow,, Phys. Fluids B, 1 (1989), 2153. doi: 10.1063/1.859081.

[36]

R. B. Dahlburg and J. M. Picone, Evolution of the Orszag-Tang vortex system in a compressible medium. II. Supersonic flow,, Phys. Fluids B, 3 (1991), 29. doi: 10.1063/1.859953.

show all references

References:
[1]

D. S. Balsara, Second order accurate schemes for magnetohydrodynamics with divergence-free reconstruction,, Astrophys. J. Suppl. Ser., 151 (2004), 149. doi: 10.1086/381377.

[2]

W. Dai and P. R. Woodward, An approximate Riemann solver for ideal magnetohydrodynamics,, J. Comp. Phys., 111 (1994), 354. doi: 10.1006/jcph.1994.1069.

[3]

D. Ryu and T. W. Jones, Numerical magnetohydrodynamics in astrophysics: Algorithm and tests for one-dimensional flow,, The Astrophys. J., 442 (1995), 228. doi: 10.1086/175437.

[4]

S. A. E. G. Falle, S. S. Komissarov and P. Joarder, A multidimensional upwind scheme for magnetohydrodynamics,, Monthly Notices of the Royal Astronomical Society, 297 (1998), 265. doi: 10.1046/j.1365-8711.1998.01506.x.

[5]

R. K. Crockett, P. Colella, R. T. Fisher, R. J. Klein and C. I. McKee, An unsplit cell-centered Godunov method for ideal MHD,, J. Comp. Phys., 203 (2005), 422. doi: 10.1016/j.jcp.2004.08.021.

[6]

D. S. Balsara, Divergence-free reconstruction of magnetic fields and WENO schemes for magnetohydrodynamics,, J. Comp. Phys., 228 (2008), 5040. doi: 10.1016/j.jcp.2009.03.038.

[7]

A. Zachary, A. Malagoli and P. Colella, A higher-order Godunov method for multidimensional ideal magnetohydrodynamics,, SIAM J. Sci. Comp., 15 (1994), 263. doi: 10.1137/0915019.

[8]

G.-S. Jiang and C.-C. Wu, A high-order WENO finite difference scheme for the equation of ideal magnetohydrodynamics,, J. Comp. Phys., 150 (1999), 561. doi: 10.1006/jcph.1999.6207.

[9]

D. S. Balsara, T. Rumpf, M. Dumbser and C.-D. Munz, Efficient, high-accuracy ADER-WENO schemes for hydrodynamics and divergence-free magnetohydrodynamics,, J. Comp. Phys., 228 (2009), 2480. doi: 10.1016/j.jcp.2008.12.003.

[10]

F. Duboc, C. Enaux, S. Jaouen, H. Jourdren and M. Wolff, High-order dimensionally split Lagrange-remap schemes for compressible hydrodynamics,, C. R. Acad. Sci. Paris, 348 (2010), 105. doi: 10.1016/j.crma.2009.12.008.

[11]

T. D. Arber, A. W. Longbottom, C. L. Gerrard and A. M. Milne, A staggered grid, Lagrangian-Eulerian remap code for 3-D MHD simulations,, J. Comp. Phys., 171 (2001), 151. doi: 10.1006/jcph.2001.6780.

[12]

S. Del Pino and H. Jourdren, Arbitrary high-order schemes for the linear advection and wave equations: Application to hydrodynamics and aeroacoustics,, C. R. Math. Acad. Sci. Paris, 342 (2006), 441. doi: 10.1016/j.crma.2006.01.013.

[13]

P. Colella, Multidimensional upwind methods for hyperbolic conservation laws,, J. Comp. Phys., 87 (1990), 171. doi: 10.1016/0021-9991(90)90233-Q.

[14]

J. U. Brackbill and D. C. Barnes, The effect of nonzero $\nabla \cdot B$ on the numerical solution of the magnetohydrodynamics equations,, J. Comp. Phys., 35 (1980), 426. doi: 10.1016/0021-9991(80)90079-0.

[15]

S. H. Brecht, J. G. Lyon, J. A. Fedder and K. Hain, A simulation study of east-west IMF effects on the magnetosphere,, Geophysical Research Letter, 8 (1981), 397. doi: 10.1029/GL008i004p00397.

[16]

C. R. DeVore, Flux-corrected transport techniques for multidimensional compressible magnetohydrodynamics,, J. Comp. Phys., 92 (1991), 142. doi: 10.1016/0021-9991(91)90295-V.

[17]

W. Dai and P. R. Woodward, On the divergence-free condition and conservation laws in numerical simulations for supersonic magnetohydrodynamic flows,, Astrophys. J., 494 (1998), 317. doi: 10.1086/305176.

[18]

D. Ryu, F. Miniati, T. W. Jones and A. Frank, A divergence-free upwind code for multidimensional magnetohydrodynamic flows,, The Astrophys. J., 509 (1998), 244. doi: 10.1086/306481.

[19]

D. S. Balsara and D. S. Spicer, Maintaining pressure positivity in magnetohydrodynamic simulations,, J. Comp. Phys., 148 (1999), 133. doi: 10.1006/jcph.1998.6108.

[20]

D. S. Balsara and D. S. Spicer, A staggered mesh algorithm using high order Godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamic simulations,, J. Comp. Phys., 149 (1999), 270. doi: 10.1006/jcph.1998.6153.

[21]

P. Londrillo and L. DelZanna, On the divergence-free condition in Godunov-type schemes for ideal magnetohydrodynamics: The upwind constrained transport method,, J. Comp. Phys., 195 (2004), 17. doi: 10.1016/j.jcp.2003.09.016.

[22]

M. Torrilhon, Locally divergence-preserving upwind finite volume schemes for magnetohydrodynamic equations,, SIAM J. Sci. Comp., 26 (2005), 1166. doi: 10.1137/S1064827503426401.

[23]

K. G. Powell, An approximate Riemann solver for MHD (that works in more than one dimension),, ICASE Report 94-24, (1994), 94.

[24]

A. Dedner, F. Kemm, D. Kröner, C.-D. Munz, T. Schnitzer and M. Wesenberg, Hyperbolic divergence cleaning for MHD equations,, J. Comp. Phys., 175 (2002), 645. doi: 10.1006/jcph.2001.6961.

[25]

G. Tóth, The $\nabla \cdot B=0$ constraint in shock-capturing magnetohydrodynamics codes,, J. Comp. Phys., 161 (2000), 605. doi: 10.1006/jcph.2000.6519.

[26]

C.-W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws,, In, 1697 (1997), 325.

[27]

E. Forest and R. D. Ruth, Fourth-order symplectic integration,, Physica D, 43 (1990), 105.

[28]

R. I. McLachlan and P. Atela, The accuracy of symplectic integrators,, Nonlinearity, 5 (1992), 541. doi: 10.1088/0951-7715/5/2/011.

[29]

S. A. Chin, Forward and non-forward symplectic integrators in solving classical dynamics problems,, Internat. J. of Comp. Math., 84 (2007), 729. doi: 10.1080/00207160701458476.

[30]

A. Cook, Artificial fluid properties for large-eddy simulation of compressible turbulent mixing,, Phys. of Fluids, 19 (2007).

[31]

N. E. L. Haugen, Hydrodynamic and hydromagnetic energy spectra from large eddy simulations,, Phys. of Fluids, 18 (2006).

[32]

M. Germano, U. Piomelli, P. Moin and W. Cabot, A dynamic subgrid-scale eddy-viscosity model,, Phys. of Fluids, 3 (1991), 1760. doi: 10.1063/1.857955.

[33]

P. Picard, Reduction and exact solutions of the ideal magnetohydrodynamics equations,, Mathemat. Phys. e-prints, (2005).

[34]

A. Orszag and C. M. Tang, Small-scale structure of two-dimensional magnetohydrodynamic turbulence,, J. Fluid. Mech., 90 (1979), 129. doi: 10.1017/S002211207900210X.

[35]

R. B. Dahlburg and J. M. Picone, Evolution of the Orszag-Tang vortex system in a compressible medium. I. Initial average subsonic flow,, Phys. Fluids B, 1 (1989), 2153. doi: 10.1063/1.859081.

[36]

R. B. Dahlburg and J. M. Picone, Evolution of the Orszag-Tang vortex system in a compressible medium. II. Supersonic flow,, Phys. Fluids B, 3 (1991), 29. doi: 10.1063/1.859953.

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