April  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), 149-184. doi: 10.1086/381377.  Google Scholar

[2]

W. Dai and P. R. Woodward, An approximate Riemann solver for ideal magnetohydrodynamics, J. Comp. Phys., 111 (1994), 354-372. doi: 10.1006/jcph.1994.1069.  Google Scholar

[3]

D. Ryu and T. W. Jones, Numerical magnetohydrodynamics in astrophysics: Algorithm and tests for one-dimensional flow, The Astrophys. J., 442 (1995), 228-258. doi: 10.1086/175437.  Google Scholar

[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), 265-277. doi: 10.1046/j.1365-8711.1998.01506.x.  Google Scholar

[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), 422-448. doi: 10.1016/j.jcp.2004.08.021.  Google Scholar

[6]

D. S. Balsara, Divergence-free reconstruction of magnetic fields and WENO schemes for magnetohydrodynamics, J. Comp. Phys., 228 (2008), 5040-5056. doi: 10.1016/j.jcp.2009.03.038.  Google Scholar

[7]

A. Zachary, A. Malagoli and P. Colella, A higher-order Godunov method for multidimensional ideal magnetohydrodynamics, SIAM J. Sci. Comp., 15 (1994), 263-284. doi: 10.1137/0915019.  Google Scholar

[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), 561-594. doi: 10.1006/jcph.1999.6207.  Google Scholar

[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), 2480-2516. doi: 10.1016/j.jcp.2008.12.003.  Google Scholar

[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), 105-110. doi: 10.1016/j.crma.2009.12.008.  Google Scholar

[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), 151-181. doi: 10.1006/jcph.2001.6780.  Google Scholar

[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), 441-446. doi: 10.1016/j.crma.2006.01.013.  Google Scholar

[13]

P. Colella, Multidimensional upwind methods for hyperbolic conservation laws, J. Comp. Phys., 87 (1990), 171-200. doi: 10.1016/0021-9991(90)90233-Q.  Google Scholar

[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), 426-430. doi: 10.1016/0021-9991(80)90079-0.  Google Scholar

[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), 397-400. doi: 10.1029/GL008i004p00397.  Google Scholar

[16]

C. R. DeVore, Flux-corrected transport techniques for multidimensional compressible magnetohydrodynamics, J. Comp. Phys., 92 (1991), 142-160. doi: 10.1016/0021-9991(91)90295-V.  Google Scholar

[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), 317-335. doi: 10.1086/305176.  Google Scholar

[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), 244-255. doi: 10.1086/306481.  Google Scholar

[19]

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

[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), 270-292. doi: 10.1006/jcph.1998.6153.  Google Scholar

[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), 17-48. doi: 10.1016/j.jcp.2003.09.016.  Google Scholar

[22]

M. Torrilhon, Locally divergence-preserving upwind finite volume schemes for magnetohydrodynamic equations, SIAM J. Sci. Comp., 26 (2005), 1166-1191. doi: 10.1137/S1064827503426401.  Google Scholar

[23]

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

[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), 645-673. doi: 10.1006/jcph.2001.6961.  Google Scholar

[25]

G. Tóth, The $\nabla \cdot B=0$ constraint in shock-capturing magnetohydrodynamics codes, J. Comp. Phys., 161 (2000), 605-652. doi: 10.1006/jcph.2000.6519.  Google Scholar

[26]

C.-W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, In "Advanced Numerical Approximation of Nonlinear Hyperbolic Equations" (Cetraro, 1997), 325-432, Lecture Notes in Math., 1697, Springer, Berlin, 1998.  Google Scholar

[27]

E. Forest and R. D. Ruth, Fourth-order symplectic integration, Physica D, 43 (1990), 105-117.  Google Scholar

[28]

R. I. McLachlan and P. Atela, The accuracy of symplectic integrators, Nonlinearity, 5 (1992), 541-562. doi: 10.1088/0951-7715/5/2/011.  Google Scholar

[29]

S. A. Chin, Forward and non-forward symplectic integrators in solving classical dynamics problems, Internat. J. of Comp. Math., 84 (2007), 729-747. doi: 10.1080/00207160701458476.  Google Scholar

[30]

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

[31]

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

[32]

M. Germano, U. Piomelli, P. Moin and W. Cabot, A dynamic subgrid-scale eddy-viscosity model, Phys. of Fluids, 3 (1991), 1760-1765. doi: 10.1063/1.857955.  Google Scholar

[33]

P. Picard, Reduction and exact solutions of the ideal magnetohydrodynamics equations, Mathemat. Phys. e-prints, 2005, arXiv:math-ph/0509048. Google Scholar

[34]

A. Orszag and C. M. Tang, Small-scale structure of two-dimensional magnetohydrodynamic turbulence, J. Fluid. Mech., 90 (1979), 129-143. doi: 10.1017/S002211207900210X.  Google Scholar

[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), 2153-2171. doi: 10.1063/1.859081.  Google Scholar

[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), 29-44. doi: 10.1063/1.859953.  Google Scholar

show all references

References:
[1]

D. S. Balsara, Second order accurate schemes for magnetohydrodynamics with divergence-free reconstruction, Astrophys. J. Suppl. Ser., 151 (2004), 149-184. doi: 10.1086/381377.  Google Scholar

[2]

W. Dai and P. R. Woodward, An approximate Riemann solver for ideal magnetohydrodynamics, J. Comp. Phys., 111 (1994), 354-372. doi: 10.1006/jcph.1994.1069.  Google Scholar

[3]

D. Ryu and T. W. Jones, Numerical magnetohydrodynamics in astrophysics: Algorithm and tests for one-dimensional flow, The Astrophys. J., 442 (1995), 228-258. doi: 10.1086/175437.  Google Scholar

[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), 265-277. doi: 10.1046/j.1365-8711.1998.01506.x.  Google Scholar

[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), 422-448. doi: 10.1016/j.jcp.2004.08.021.  Google Scholar

[6]

D. S. Balsara, Divergence-free reconstruction of magnetic fields and WENO schemes for magnetohydrodynamics, J. Comp. Phys., 228 (2008), 5040-5056. doi: 10.1016/j.jcp.2009.03.038.  Google Scholar

[7]

A. Zachary, A. Malagoli and P. Colella, A higher-order Godunov method for multidimensional ideal magnetohydrodynamics, SIAM J. Sci. Comp., 15 (1994), 263-284. doi: 10.1137/0915019.  Google Scholar

[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), 561-594. doi: 10.1006/jcph.1999.6207.  Google Scholar

[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), 2480-2516. doi: 10.1016/j.jcp.2008.12.003.  Google Scholar

[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), 105-110. doi: 10.1016/j.crma.2009.12.008.  Google Scholar

[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), 151-181. doi: 10.1006/jcph.2001.6780.  Google Scholar

[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), 441-446. doi: 10.1016/j.crma.2006.01.013.  Google Scholar

[13]

P. Colella, Multidimensional upwind methods for hyperbolic conservation laws, J. Comp. Phys., 87 (1990), 171-200. doi: 10.1016/0021-9991(90)90233-Q.  Google Scholar

[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), 426-430. doi: 10.1016/0021-9991(80)90079-0.  Google Scholar

[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), 397-400. doi: 10.1029/GL008i004p00397.  Google Scholar

[16]

C. R. DeVore, Flux-corrected transport techniques for multidimensional compressible magnetohydrodynamics, J. Comp. Phys., 92 (1991), 142-160. doi: 10.1016/0021-9991(91)90295-V.  Google Scholar

[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), 317-335. doi: 10.1086/305176.  Google Scholar

[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), 244-255. doi: 10.1086/306481.  Google Scholar

[19]

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

[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), 270-292. doi: 10.1006/jcph.1998.6153.  Google Scholar

[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), 17-48. doi: 10.1016/j.jcp.2003.09.016.  Google Scholar

[22]

M. Torrilhon, Locally divergence-preserving upwind finite volume schemes for magnetohydrodynamic equations, SIAM J. Sci. Comp., 26 (2005), 1166-1191. doi: 10.1137/S1064827503426401.  Google Scholar

[23]

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

[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), 645-673. doi: 10.1006/jcph.2001.6961.  Google Scholar

[25]

G. Tóth, The $\nabla \cdot B=0$ constraint in shock-capturing magnetohydrodynamics codes, J. Comp. Phys., 161 (2000), 605-652. doi: 10.1006/jcph.2000.6519.  Google Scholar

[26]

C.-W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, In "Advanced Numerical Approximation of Nonlinear Hyperbolic Equations" (Cetraro, 1997), 325-432, Lecture Notes in Math., 1697, Springer, Berlin, 1998.  Google Scholar

[27]

E. Forest and R. D. Ruth, Fourth-order symplectic integration, Physica D, 43 (1990), 105-117.  Google Scholar

[28]

R. I. McLachlan and P. Atela, The accuracy of symplectic integrators, Nonlinearity, 5 (1992), 541-562. doi: 10.1088/0951-7715/5/2/011.  Google Scholar

[29]

S. A. Chin, Forward and non-forward symplectic integrators in solving classical dynamics problems, Internat. J. of Comp. Math., 84 (2007), 729-747. doi: 10.1080/00207160701458476.  Google Scholar

[30]

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

[31]

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

[32]

M. Germano, U. Piomelli, P. Moin and W. Cabot, A dynamic subgrid-scale eddy-viscosity model, Phys. of Fluids, 3 (1991), 1760-1765. doi: 10.1063/1.857955.  Google Scholar

[33]

P. Picard, Reduction and exact solutions of the ideal magnetohydrodynamics equations, Mathemat. Phys. e-prints, 2005, arXiv:math-ph/0509048. Google Scholar

[34]

A. Orszag and C. M. Tang, Small-scale structure of two-dimensional magnetohydrodynamic turbulence, J. Fluid. Mech., 90 (1979), 129-143. doi: 10.1017/S002211207900210X.  Google Scholar

[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), 2153-2171. doi: 10.1063/1.859081.  Google Scholar

[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), 29-44. doi: 10.1063/1.859953.  Google Scholar

[1]

Kai Jiang, Wei Si. High-order energy stable schemes of incommensurate phase-field crystal model. Electronic Research Archive, 2020, 28 (2) : 1077-1093. doi: 10.3934/era.2020059

[2]

Abdollah Borhanifar, Maria Alessandra Ragusa, Sohrab Valizadeh. High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5495-5508. doi: 10.3934/dcdsb.2020355

[3]

Lela Dorel. Glucose level regulation via integral high-order sliding modes. Mathematical Biosciences & Engineering, 2011, 8 (2) : 549-560. doi: 10.3934/mbe.2011.8.549

[4]

Guoshan Zhang, Peizhao Yu. Lyapunov method for stability of descriptor second-order and high-order systems. Journal of Industrial & Management Optimization, 2018, 14 (2) : 673-686. doi: 10.3934/jimo.2017068

[5]

Yuezheng Gong, Jiaquan Gao, Yushun Wang. High order Gauss-Seidel schemes for charged particle dynamics. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 573-585. doi: 10.3934/dcdsb.2018034

[6]

Abdelwahab Bensouilah, Sahbi Keraani. Smoothing property for the $ L^2 $-critical high-order NLS Ⅱ. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2961-2976. doi: 10.3934/dcds.2019123

[7]

Kolade M. Owolabi, Abdon Atangana. High-order solvers for space-fractional differential equations with Riesz derivative. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 567-590. doi: 10.3934/dcdss.2019037

[8]

Marc Bonnet. Inverse acoustic scattering using high-order small-inclusion expansion of misfit function. Inverse Problems & Imaging, 2018, 12 (4) : 921-953. doi: 10.3934/ipi.2018039

[9]

Raymond H. Chan, Haixia Liang, Suhua Wei, Mila Nikolova, Xue-Cheng Tai. High-order total variation regularization approach for axially symmetric object tomography from a single radiograph. Inverse Problems & Imaging, 2015, 9 (1) : 55-77. doi: 10.3934/ipi.2015.9.55

[10]

Shan Jiang, Li Liang, Meiling Sun, Fang Su. Uniform high-order convergence of multiscale finite element computation on a graded recursion for singular perturbation. Electronic Research Archive, 2020, 28 (2) : 935-949. doi: 10.3934/era.2020049

[11]

Tarek Saanouni. Global well-posedness of some high-order semilinear wave and Schrödinger type equations with exponential nonlinearity. Communications on Pure & Applied Analysis, 2014, 13 (1) : 273-291. doi: 10.3934/cpaa.2014.13.273

[12]

Zheng Sun, José A. Carrillo, Chi-Wang Shu. An entropy stable high-order discontinuous Galerkin method for cross-diffusion gradient flow systems. Kinetic & Related Models, 2019, 12 (4) : 885-908. doi: 10.3934/krm.2019033

[13]

Phillip Colella. High-order finite-volume methods on locally-structured grids. Discrete & Continuous Dynamical Systems, 2016, 36 (8) : 4247-4270. doi: 10.3934/dcds.2016.36.4247

[14]

Andrey B. Muravnik. On the Cauchy problem for differential-difference parabolic equations with high-order nonlocal terms of general kind. Discrete & Continuous Dynamical Systems, 2006, 16 (3) : 541-561. doi: 10.3934/dcds.2006.16.541

[15]

Roger P. de Moura, Ailton C. Nascimento, Gleison N. Santos. On the stabilization for the high-order Kadomtsev-Petviashvili and the Zakharov-Kuznetsov equations with localized damping. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021022

[16]

Luca Calatroni, Bertram Düring, Carola-Bibiane Schönlieb. ADI splitting schemes for a fourth-order nonlinear partial differential equation from image processing. Discrete & Continuous Dynamical Systems, 2014, 34 (3) : 931-957. doi: 10.3934/dcds.2014.34.931

[17]

Cheng Wang, Jian-Guo Liu. Positivity property of second-order flux-splitting schemes for the compressible Euler equations. Discrete & Continuous Dynamical Systems - B, 2003, 3 (2) : 201-228. doi: 10.3934/dcdsb.2003.3.201

[18]

Ahmed El Kaimbillah, Oussama Bourihane, Bouazza Braikat, Mohammad Jamal, Foudil Mohri, Noureddine Damil. Efficient high-order implicit solvers for the dynamic of thin-walled beams with open cross section under external arbitrary loadings. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1685-1708. doi: 10.3934/dcdss.2019113

[19]

Florian Schneider, Jochen Kall, Graham Alldredge. A realizability-preserving high-order kinetic scheme using WENO reconstruction for entropy-based moment closures of linear kinetic equations in slab geometry. Kinetic & Related Models, 2016, 9 (1) : 193-215. doi: 10.3934/krm.2016.9.193

[20]

Wenqiang Zhao, Yijin Zhang. High-order Wong-Zakai approximations for non-autonomous stochastic $ p $-Laplacian equations on $ \mathbb{R}^N $. Communications on Pure & Applied Analysis, 2021, 20 (1) : 243-280. doi: 10.3934/cpaa.2020265

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (187)
  • HTML views (0)
  • Cited by (3)

[Back to Top]