×

zbMATH — the first resource for mathematics

Hybrid block-AMR in Cartesian and curvilinear coordinates: MHD applications. (English) Zbl 1310.76133
Summary: We present a novel, hybrid block-adaptive scheme for use in solving sets of near-conservation laws in general orthogonal coordinate systems. The adaptive mesh refinement (AMR) scheme is block-based, i.e. individual grids have a pre-fixed number of grid cells, and is implemented for any-dimensionality \(D\). Its ‘hybrid’ character relaxes the common approach where a block that needs refinement triggers \(2^{D}\) subblocks when grids are refined by a factor of 2. This introduces ‘incomplete families’ in the grid hierarchy, but approaches the optimal fit to developing flow features inherent in the original patch-based AMR strategy. Our hybrid block-AMR approach is compared with a patch-based AMR one, both exploiting OpenMP parallelism. The implementation is able to handle general curvilinear coordinates, for which restriction and prolongation formulae are presented along with boundary treatments at ‘singular’ boundaries. Demonstrative examples cover hydro- and magnetohydrodynamic (MHD) applications, including tests on a 2D polar grid, a 2.5D spherical and a 3D cylindrical configuration. The applications cover classical up to relativistic MHD simulations, of particular relevance for astrophysical magnetized jet and stellar wind studies.

MSC:
76M25 Other numerical methods (fluid mechanics) (MSC2010)
76W05 Magnetohydrodynamics and electrohydrodynamics
Software:
Racoon; NIRVANA
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Balsara, D.S., Second-order accurate schemes for magnetohydrodynamics with divergence-free reconstruction, Astrophys. J. suppl. ser., 151, 149, (2004)
[2] Baty, H.; Keppens, R.; Comte, P., The two-dimensional magnetohydrodynamic kelvin – helmholtz instability: compressibility and large-scale coalescence effects, Phys. plasmas., 10, 4661, (2003)
[3] Bell, J.; Berger, M.; Saltzman, J.; Welcome, M., Three-dimensional adaptive mesh refinement for hyperbolic conservation laws, SIAM J. sci. comp., 15, 127, (1994) · Zbl 0793.65072
[4] Berger, M.J., Data structures for adaptive grid generation, SIAM J. sci. stat. comput., 7, 904, (1986) · Zbl 0625.65116
[5] Berger, M.J.; Colella, P., Local adaptive mesh refinement for shock hydrodynamics, J. comput. phys., 82, 64, (1989) · Zbl 0665.76070
[6] Berger, M.J.; Rigoutsos, I., An algorithm for point clustering and grid generation, IEEE trans. systems, man cybernetics, 21, 1278, (1991)
[7] J. Bergmans, R. Keppens, D.E.A. van Odyck, A. Achterberg, Simulations of relativistic astrophysical flows, in: T. Plewa, T. Linde, V.G. Weirs (Eds.), Adaptive Mesh Refinement - Theory and Applications, Lect. Not. Comp. Sci. Eng., vol. 41, 2005, pp. 223. · Zbl 1065.85502
[8] Berkeley Lab AMR homepage. <http://seesar.lbl.gov/>.
[9] Calder, A.C.; Fryxell, B.; Plewa, T.; Rosner, R.; Dursi, L.J.; Weirs, V.G.; Dupont, T.; Robey, H.F.; Kane, J.O.; Remington, B.A.; Drake, R.P.; Dimonte, G.; Zingale, M.; Timmes, F.X.; Olson, K.; Ricker, P.; MacNeice, P.; Tufo, H.M., On validating an astrophysical simulation code, Astrophys. J. suppl. ser., 143, 201, (2002)
[10] Casse, F.; Marcowith, A., Astroparticle yield and transport from extragalactic jet terminal shocks, Astroparticle phys., 23, 31, (2005)
[11] van Dam, A.; Zegeling, P.A., A robust moving mesh finite volume method applied to 1D hyperbolic conservation laws from magnetohydrodynamics, J. comput. phys., 216, 526, (2006) · Zbl 1102.35358
[12] De Sterck, H.; Csik, A.; Vanden Abeele, D.; Poedts, S.; Deconinck, H., Stationary two-dimensional magnetohydrodynamic flows with shocks: characteristic analysis and grid convergence study, J. comput. phys., 166, 28, (2001) · Zbl 1029.76034
[13] De Zeeuw, D.; Powell, K.G., An adaptive refined Cartesian mesh solver for the Euler equations, J. comput. phys., 104, 56, (1993) · Zbl 0766.76066
[14] Dreher, J.; Grauer, R., Racoon: a parallel mesh-adaptive framework for hyperbolic conservation laws, Par. comp., 31, 913, (2005)
[15] Evans, C.R.; Hawley, J.F., Simulation of magnetohydrodynamic flows – a constrained transport method, Astrophys. J., 332, 659, (1988)
[16] FLASH user guide. <http://flash.uchicago.edu>.
[17] Friedel, H.; Grauer, R.; Marliani, C., Adaptive mesh refinement for singular current sheets in incompressible magnetohydrodynamic flows, J. comput. phys., 134, 190, (1997) · Zbl 0879.76079
[18] Garcia, A.L.; Bell, J.B.; Crutchfield, W.Y.; Alder, B.J., Adaptive mesh and algorithm refinement using direct simulation Monte Carlo, J. comput. phys., 154, 134, (1999) · Zbl 0954.76075
[19] T.I. Gombosi, K.G. Powell, D.L. de Zeeuw, C.R. Clauer, K.C. Hansen, W.B. Manchester, A.J. Ridley, I.I. Roussev, I.V. Sokolov, Q.F. Stout, G. Tóth, Solution-adaptive magnetohydrodynamics for space plasmas: sun-to-earth simulations, Comput. Sci. Eng. March/April issue, 14 (2004).
[20] Groth, C.P.T.; De Zeeuw, D.L.; Gombosi, T.I.; Powell, K.G., Global three-dimensional MHD simulation of a space weather event: CME formation, interplanetary propagation, and interaction with the magnetosphere, J. geophys. res., 105, 25053, (2000)
[21] Harten, A., High resolution schemes for hyperbolic conservation laws, J. comput. phys., 49, 357, (1983) · Zbl 0565.65050
[22] Jacobs, C.; Poedts, S.; van der Holst, B., The effect of the solar wind on CME triggering by magnetic foot point shearing, Astron. astrophys., 450, 793, (2006)
[23] R. Keppens, Z. Meliani, Grid-adaptive simulations of relativistic flows, in: Proceedings of ICCFD4, July 2006, Ghent, Belgium.
[24] Leismann, T.; Antón, L.; Aloy, M.A.; Müller, E.; Martí, J.M.; Miralles, J.A.; Ibáñez, J.M., Relativistic MHD simulations of extragalactic jets, Astron. astrophys., 436, 503, (2005)
[25] Li, S.; Li, H., A novel approach of divergence-free reconstruction for adaptive mesh refinement, J. comput. phys., 199, 1, (2004) · Zbl 1054.65121
[26] Marder, B., A method for incorportaing gauss’ law into electromagnetic PIC codes, J. comput. phys., 68, 48, (1987) · Zbl 0603.65079
[27] Mignone, A.; Massaglia, S.; Bodo, G., Relativistic MHD simulations of jets with toroidal magnetic fields, Space sci. rev., 121, 21, (2005)
[28] Keppens, R.; Nool, M.; Tóth, G.; Goedbloed, J.P., Adaptive mesh refinement for conservative systems: multi-dimensional efficiency evaluation, Comput. phys. commun., 153, 317, (2003) · Zbl 1196.76055
[29] Manchester, W.B.; Gombosi, T.I.; Roussev, I.; Ridley, A.; de Zeeuw, D.L.; Sokolov, I.V.; Powell, K.G.; Tóth, G., Modeling a space weather event from the Sun to the Earth: CME generation and interplanetary propagation, J. geophys. res., 109, A02107, (2004)
[30] Parker, E.N., Dynamics of the interplanetary gas and magnetic fields, Astrophys. J., 128, 664, (1958)
[31] Quirk, J.J., A contribution to the great Riemann solver debate, Int. J. numer. methods fl., 18, 555, (1994) · Zbl 0794.76061
[32] Rusanov, V.V., The calculation of the interaction of non-stationary shock waves and obstacles, USSR comput. math. math. phys., 1, 304, (1961)
[33] SAMRAI Project at LLNL. <http://www.llnl.gov/casc/SAMRAI>.
[34] Torrilhon, M., Non-uniform convergence of finite volume schemes for Riemann problems of ideal magnetohydrodynamics, J. comput. phys., 192, 73, (2003) · Zbl 1032.76721
[35] Torrilhon, M., Uniqueness conditions for Riemann problems of ideal magnetohydrodynamics, J. plasma phys., 69, 253, (2003)
[36] Torrillhon, M.; Balsara, D.S., High order WENO schemes: investigations on non-uniform convergence for MHD Riemann problems, J. comput. phys., 201, 586, (2004) · Zbl 1076.76050
[37] Tóth, G., The LASY preprocessor and its application to general multi-dimensional codes, J. comput. phys., 138, 981, (1997) · Zbl 0903.76077
[38] Tóth, G., A general code for modeling MHD flows on parallel computers: versatile advection code, Astrophys. lett. commun., 34, 245, (1996), See
[39] Tóth, G.; Odstrčil, D., Comparison of some flux corrected transport and total variation diminishing numerical schemes for hydrodynamic and magnetohydrodynamic problems, J. comput. phys., 128, 82, (1996) · Zbl 0860.76061
[40] Woodward, P.R.; Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. comput. phys., 54, 174, (1984)
[41] H.C. Yee, A class of high-resolution explicit and implicit shock-capturing methods, NASA TM-101088, 1989.
[42] Ziegler, U., Self-gravitational adaptive mesh magnetohydrodynamics with the NIRVANA code, Astron. astrophys., 435, 385, (2005)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.