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A second-order unsplit Godunov scheme for cell-centered MHD: the CTU-GLM scheme. (English) Zbl 1303.76142
Summary: We assess the validity of a single step Godunov scheme for the solution of the magnetohydrodynamics equations in more than one dimension. The scheme is second-order accurate and the temporal discretization is based on the dimensionally unsplit Corner Transport Upwind (CTU) method of Colella. The proposed scheme employs a cell-centered representation of the primary fluid variables (including magnetic field) and conserves mass, momentum, magnetic induction and energy. A variant of the scheme, which breaks momentum and energy conservation, is also considered. Divergence errors are transported out of the domain and damped using the mixed hyperbolic/parabolic divergence cleaning technique by A. Dedner et al. [in: Hyperbolic problems: Theory, numerics, applications. Proceedings of the ninth international conference on hyperbolic problems, 2002. Berlin: Springer, 509–518 (2003; Zbl 1061.76036)]. The strength and accuracy of the scheme are verified by a direct comparison with the eight-wave formulation (also employing a cell-centered representation) and with the popular constrained transport method, where magnetic field components retain a staggered collocation inside the computational cell. Results obtained from two- and three-dimensional test problems indicate that the newly proposed scheme is robust, accurate and competitive with recent implementations of the constrained transport method while being considerably easier to implement in existing hydro codes.

76W05 Magnetohydrodynamics and electrohydrodynamics
76M12 Finite volume methods applied to problems in fluid mechanics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
Full Text: DOI
[1] Artebrant, A.; Torrilhon, M., Increasing the accuracy in locally divergence-preserving finite-volume schemes for MHD, J. comput. phys., 227, 3405-3427, (2008) · Zbl 1329.76197
[2] Balsara, D.S.; Spicer, D.S., A staggered mesh algorithm using high order Godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamics simulations, J. comput. phys., 149, 270, (1999) · Zbl 0936.76051
[3] Balsara, D.S., Second-order-accurate schemes for magnetohydrodynamics with divergence-free reconstruction, Astrophys. J. suppl., 151, 149, (2004)
[4] Balsara, D.S.; Kim, J., A comparison between divergence-cleaning and staggered-mesh formulations for numerical magnetohydrodynamics, Apj, 602, 1079, (2004)
[5] Banks, J.W.; Aslam, T.; Rider, W.J., On sub-linear convergence for linearly degenerate waves in capturing schemes, J. comput. phys., 227, 6985-7002, (2008) · Zbl 1145.65061
[6] Brackbill, J.U.; Barnes, D.C., The effect of nonzero \(\nabla \cdot B\) on the numerical solution of the magnetohydrodynamics equations, J. comput. phys., 35, 426, (1980) · Zbl 0429.76079
[7] Cargo, P.; Gallice, G., Roe matrices for ideal MHD and systematic construction of roe matrices for systems of conservation laws, J. comput. phys., 136, 446, (1997) · Zbl 0919.76053
[8] Colella, P., Multidimensional upwind methods for hyperbolic conservation laws, J. comput. phys., 87, 171, (1990) · Zbl 0694.65041
[9] Crockett, R.K.; Colella, P.; Fisher, R.T.; Klein, R.I.; McKee, C.F., An unsplit, cell-centered Godunov method for ideal MHD, J. comput. phys., 203, 422, (2005) · Zbl 1143.76599
[10] Dai, W.; Woodward, P.R., An approximate Riemann solver for ideal magnetohydrodynamics, J. comput. phys., 111, 372, (1994) · Zbl 0797.76052
[11] Dedner, A.; Kemm, F.; Kröner, D.; Munz, C.D.; Schnitzer, T.; Wesenberg, M., Hyperbolic divergence cleaning for the MHD equations, J. comput. phys., 175, 645-673, (2002) · Zbl 1059.76040
[12] Evans, C.R.; Hawley, J.F., Simulation of magnetohydrodynamics flows – a constrained transport method, Apj, 332, 659, (1988)
[13] Fromang, S.; Hennebelle, P.; Teyssier, R., A high order Godunov scheme with constrained transport and adaptive mesh refinement for astrophysical magnetohydrodynamics, Astron. astrophys., 457, 371, (2006)
[14] Gardiner, T.; Stone, J., An unsplit Godunov method for ideal MHD via constrained transport, J. comput. phys., 205, 509, (2005) · Zbl 1087.76536
[15] Gardiner, T.; Stone, J., An unsplit Godunov method for ideal MHD via constrained transport in three dimensions, J. comput. phys., 227, 4123, (2008) · Zbl 1317.76057
[16] Lee, D.; Deane, A.E., An unsplit staggered mesh scheme for multidimensional magnetohydrodynamics, J. comput. phys., 228, 952, (2009) · Zbl 1330.76093
[17] Londrillo, P.; Del Zanna, L., On the divergence-free condition in Godunov-type schemes for ideal magnetohydrodynamics: the upwind constrained transport method, J. comput. phys., 195, 17, (2004) · Zbl 1087.76074
[18] S. Massaglia, G. Bodo, A. Mignone, P. Rossi, Jets from young stars III: Numerical MHD and instabilities, Lecture Notes in Physics, vol. 754, Springer-Verlag, Berlin/Heidelberg, 2008, ISBN: 978-3-540-76966-8. · Zbl 1151.85002
[19] Mignone, A.; Bodo, G.; Massaglia, S., PLUTO: a numerical code for computational astrophysics, Apjs, 170, 228, (2007)
[20] K.G. Powell, An Approximate Riemann Solver for Magnetohydrodynamics (That Works in More than One Dimension), ICASE-Report 94-24 (NASA CR-194902), NASA Langley Research Center, Hampton, VA 23681-0001, 8 April 1994.
[21] Powell, K.G.; Roe, P.L.; Linde, T.J.; Gombosi, T.I.; De Zeeuw, D.L., A solution-adaptive upwind scheme for ideal magnetohydrodynamics, J. comput. phys., 154, 284, (1999) · Zbl 0952.76045
[22] Rossmanith, J.A., An unstaggered, high-resolution constrained transport method for magnetohydrodynamic flows, SIAM J. sci. comput., 28, 1766-1797, (2006) · Zbl 1344.76092
[23] Ryu, D.; Jones, T.W.; Frank, A., Numerical magnetohydrodynamics in astrophysics: algorithm and tests for multidimensional flows, Apj, 452, 785, (1995)
[24] Ryu, D.; Miniati, F.; Jones, T.W.; Frank, A., A divergence-free upwind code for multidimensional magnetohydrodynamic flows, Apj, 509, 244, (1998)
[25] Tóth, G., The \(\nabla \cdot B = 0\) constraint in shock-capturing magnetohydrodynamics codes, J. comput. phys., 161, 605, (2000) · Zbl 0980.76051
[26] Torrilhon, M., Locally divergence-preserving upwind finite volume schemes for magnetohydrodynamics equations, SIAM J. sci. comput., 26, 1166, (2005) · Zbl 1149.76693
[27] Zachary, A.L.; Malagoli, A.; Colella, P., A high-order Godunov method for multidimensional ideal MHD, SIAM J. sci. comput., 15, 263, (1994) · Zbl 0797.76063
[28] Ziegler, U., A central-constrained transport scheme for ideal magnetohydrodynamics, J. comput. phys., 196, 393, (2004) · Zbl 1115.76427
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