An ideal compressible magnetohydrodynamic solver with parallel block-structured adaptive mesh refinement. (English) Zbl 1410.76252

Summary: We present an adaptive parallel solver for the numerical simulation of ideal magnetohydrodynamics in two and three space dimensions. The discretisation uses a finite volume scheme based on a Cartesian mesh and an explicit compact Runge-Kutta scheme for time integration. Numerically, a generalized Lagrangian multiplier approach with a mixed hyperbolic-parabolic correction is used to guarantee a control on the incompressibility of the magnetic field. We implement the solver in the AMROC (adaptive mesh refinement in object-oriented C++) framework that uses a structured adaptive mesh refinement (SAMR) method discretisation-independent and is fully parallelised for distributed memory systems. Moreover, AMROC is a modular framework providing manageability, extensibility and efficiency. In this paper, we give an overview of the ideal magnetohydrodynamics solver developed in this framework and its capabilities. We also include an example of this solver’s verification with other codes and its numerical and computational performance.


76M12 Finite volume methods applied to problems in fluid mechanics
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
76W05 Magnetohydrodynamics and electrohydrodynamics


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[1] Tóth, G.; van der Holst, B.; Sokolov, I. V.; De Zeeuw, D. L.; Gombosi, T. I.; Fang, F., Adaptive numerical algorithms in space weather modeling, J Comput Phys, 231, 3, 870-903, (2012)
[2] Bittencourt, J., Fundamentals of plasma physics, (2004), Springer · Zbl 1084.76001
[3] Hopkins, P. F., A constrained-gradient method to control divergence errors in numerical MHD, Mon Notices R Astron Soc, 462, 11, 576-587, (2016)
[4] Gomes, A. K.F.; Domingues, M. O.; Schneider, K.; Mendes, O.; Deiterding, R., An adaptive multiresolution method for ideal magnetohydrodynamics using divergence cleaning with parabolic-hyperbolic correction, Appl Numer Math, 95, 199-213, (2015) · Zbl 1320.76080
[5] Domingues, M. O.; Gomes, A. K.F.; Gomes, S. M.; Mendes, O.; Di Pierro, B.; Schneider, K., Extended generalized Lagrangian multipliers for magnetohydrodynamics using adaptive multiresolution methods, ESAIM: Proc, 43, 95-107, (2013) · Zbl 1329.76202
[6] Majda, A., Compressible fluid flow and systems of conservation laws in several space variables, (1984), Springer · Zbl 0537.76001
[7] Brackbill, J. U.; Barnes, D. C., The effect of nonzero \(\nabla \cdot \mathbf{B}\) on the numerical solution of the magnetohysdrodynamic equations, J Comput Phys, 35, 426-430, (1980) · Zbl 0429.76079
[8] Dedner, A.; Kemm, F.; Kröner, D.; Munz, C.-D.; Schnitzer, T.; Wesenberg, M., Hyperbolic divergence cleaning for the MHD equations, J Comput Phys, 175, 2, 645-673, (2002) · Zbl 1059.76040
[9] Mignone, A.; Tzeferacos, P., A second-order unsplit Godunov scheme for cell-centered MHD: the CTU-GLM scheme, J Comput Phys, 229, 6, 2117-2138, (2010) · Zbl 1303.76142
[10] Leveque, R. J., Finite volume methods for hyperbolic systems, (2002), Cambridge University Press · Zbl 1010.65040
[11] Miyoshi, T.; Kusano, K. A., A multi-state HLL approximate Riemann solver for ideal magnetohydrodynamics, J Comput Phys, 208, 315-344, (2005) · Zbl 1114.76378
[12] Toro, E. F., Riemann solvers and numerical methods for fluid dynamics, (1999), Springer · Zbl 0923.76004
[13] Berger, M.; Colella, P., Local adaptive mesh refinement for shock hydrodynamics, J Comput Phys, 82, 64-84, (1989) · Zbl 0665.76070
[14] Deiterding, R., Block-structured adaptive mesh refinement - theory, implementation and application, ESAIM: Proc, 34, 97-150, (2011) · Zbl 1302.65220
[15] Deiterding, R., Parallel adaptive simulation of multi-dimensional detonation structures, (2003), Brandenburgische Technische Universität Cottbus, Ph.D. thesis
[16] Deiterding, R., Construction and application of an AMR algorithm for distributed memory computers, (Plewa, T.; Linde, T.; Weirs, V. G., Adaptive mesh refinement - theory and applications, (2005), Springer), 361-372 · Zbl 1065.65114
[17] Bell, J.; Berger, M.; Saltzmann, J.; Welcome, M., Three-dimensional adaptive mesh refinement for hyperbolic conservation laws, SIAM J Sci Comput, 15, 127-138, (1994) · Zbl 0793.65072
[18] Frank, A.; Jones, T. W.; Ryu, D.; Gaalaas, J. B., The magnetohydrodynamic Kelvin-Helmholtz instability: a two-dimensional numerical study, Astrophys J, 460, 777-793, (1996)
[19] Orszag, S. A.; Tang, C.-M., Small-scale structure of two-dimensional magnetohydrodynamic turbulence, J Fluid Mech, 90, 1, 129-143, (1979)
[20] Ryu, D.; Miniati, F.; Jones, T.; Frank, A., A divergence-free upwind code for multidimensional magnetohydrodynamic flows, Astrophys J, 509, 1, 244-255, (1998)
[21] Dai, W.; Woodward, P. R., A simple finite difference scheme for multidimensional magnetohydrodynamical equations, J Comput Phys, 142, 2, 331-369, (1998) · Zbl 0932.76048
[22] Londrillo, P.; Del Zanna, L., High-order upwind schemes for multidimensional magnetohydrodynamics, Astrophys J, 530, 1, 508-524, (2000)
[23] Picone, J. M.; Dahlburg, R. B., Evolution of the orszag-Tang vortex system in a compressible medium. II. supersonic flow, Phys Fluids B, 3, 1, 29-44, (1991)
[24] Helzel, C.; Rossmanith, J. A.; Taetz, B., An unstaggered constrained transport method for the 3D ideal magnetohydrodynamic equations, J Comput Phys, 230, 10, 3803-3829, (2011) · Zbl 1369.76061
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