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Divergence-free MHD simulations with the HERACLES code. (English. French summary) Zbl 1329.76401

Summary: Numerical simulations of the magnetohydrodynamics (MHD) equations have played a significant role in plasma research over the years. The need of obtaining physical and stable solutions to these equations has led to the development of several schemes, all requiring to satisfy and preserve the divergence constraint of the magnetic field numerically. In this paper, we aim to show the importance of maintaining this constraint numerically. We investigate in particular the hyperbolic divergence cleaning technique applied to the ideal MHD equations on a collocated grid and compare it to the constrained transport technique that uses a staggered grid to maintain the property. The methods are implemented in the software HERACLES and several numerical tests are presented, where the robustness and accuracy of the different schemes can be directly compared.

MSC:

76W05 Magnetohydrodynamics and electrohydrodynamics
76M20 Finite difference methods applied to problems in fluid mechanics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65Y15 Packaged methods for numerical algorithms

Software:

HE-E1GODF; HERACLES
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Full Text: DOI

References:

[1] D. S. Balsara and D. S. Spicer. A staggered mesh algorithm using high order Godunov uxes to ensure solenoidal magnetic elds in magnetohydrodynamic simulations. Journal of Computational Physics, 149(2):270292, 1999. · Zbl 0936.76051
[2] T. Barth. On the role of involutions in the discontinuous Galerkin discretization of Maxwell and magnetohydrodynamic systems. In Compatible Spatial Discretizations, volume 142 of The IMA Volumes in Mathematics and its Applications, pages 6988. Springer New York, 2006. · Zbl 1135.78008
[3] J. U. Brackbill and D. C. Barnes. The eect of nonzero {\(\cdot\)}B on the numerical solution of the magnetohydrodynamic equations.Journal of Computational Physics, 35(3):426430, 1980. · Zbl 0429.76079
[4] W. Dai and P. R. Woodward. On the divergence-free condition and conservation laws in numerical simulations for supersonic magnetohydrodynamical ows. The Astrophysical Journal, 494(1):317, 1998.
[5] A. Dedner, F. Kemm, D. Kröner, C.-D. Munz, T. Schnitzer, and M. Wesenberg. Hyperbolic divergence cleaning for the MHD equations. Journal of Computational Physics, 175(2):645673, 2002. · Zbl 1059.76040
[6] C. R. Evans and J. F. Hawley. Simulation of magnetohydrodynamic ows: a constrained transport method. Astrophysical Journal, Part 1, 332:659677, September 1988.
[7] S. Fromang, P. Hennebelle, and R. Teyssier. A high order Godunov scheme with constrained transport and adaptive mesh renement for astrophysical MHD. A&A, 457:371384, 2006.
[8] C. Helzel, J. A. Rossmanith, and B. Taetz. An unstaggered constrained transport method for the 3D ideal magnetohydrodynamic equations. Journal of Computational Physics, 230(10):38033829, 2011. · Zbl 1369.76061
[9] P. Londrillo and L. Del Zanna. High-order upwind schemes for multidimensional magnetohydrodynamics. The Astrophysical Journal, 530(1):508, 2000. · Zbl 1087.76074
[10] A. Mignone, P. Tzeferacos, and G. Bodo. High-order conservative nite dierence GLMMHD schemes for cell-centered MHD. Journal of Computational Physics, 229(17):58965920, 2010. · Zbl 1425.76305
[11] T. Miyoshi and K. Kusano. A multi-state HLL approximate Riemann solver for ideal magnetohydrodynamics. Journal of Computational Physics, 208(1):315344, 2005. · Zbl 1114.76378
[12] K. G. Powell. An approximate Riemann solver for magnetohydrodynamics (that works in more than one dimension). Technical Report 94-24, ICASE, Langley, VA, 1994.
[13] E. F. Toro. Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction. Springer, 2009. · Zbl 1227.76006
[14] G. Tóth. The{\(\cdot\)} B = 0 constraint in shock-capturing magnetohydrodynamics codes. Journal of Computational Physics,161(2):605652, 2000. · Zbl 0980.76051
[15] K. S. Yee. Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media. IEEE Trans. Antennas and Propagation, pages 302307, 1966. · Zbl 1155.78304
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