×

Jacobian-free approximate solvers for hyperbolic systems: application to relativistic magnetohydrodynamics. (English) Zbl 1411.65119

Summary: We present recent advances in PVM (Polynomial Viscosity Matrix) methods based on internal approximations to the absolute value function, and compare them with Chebyshev-based PVM solvers. These solvers only require a bound on the maximum wave speed, so no spectral decomposition is needed. Another important feature of the proposed methods is that they are suitable to be written in Jacobian-free form, in which only evaluations of the physical flux are used. This is particularly interesting when considering systems for which the Jacobians involve complex expressions, e.g., the relativistic magnetohydrodynamics (RMHD) equations. On the other hand, the proposed Jacobian-free solvers have also been extended to the case of approximate DOT (Dumbser-Osher-Toro) methods, which can be regarded as simple and efficient approximations to the classical Osher-Solomon method, sharing most of it interesting features and being applicable to general hyperbolic systems. To test the properties of our schemes a number of numerical experiments involving the RMHD equations are presented, both in one and two dimensions. The obtained results are in good agreement with those found in the literature and show that our schemes are robust and accurate, running stable under a satisfactory time step restriction. It is worth emphasizing that, although this work focuses on RMHD, the proposed schemes are suitable to be applied to general hyperbolic systems.

MSC:

65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
76M12 Finite volume methods applied to problems in fluid mechanics
76Y05 Quantum hydrodynamics and relativistic hydrodynamics
76W05 Magnetohydrodynamics and electrohydrodynamics
35L65 Hyperbolic conservation laws
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abbott, B. P., Phys. Rev. Lett., 116, 061102 (2016)
[2] Wilson, J. R., Astrophys. J., 173, 431-438 (1972)
[3] Wilson, J. R., (Smarr, L. L., Sources of Gravitational Radiation (1979), Cambridge University Press: Cambridge University Press Cambridge, UK), 423-446 · Zbl 0407.00016
[4] Martí, J. M.; Müller, E., Living Rev. Relativ., 6, 7 (2003) · Zbl 1068.83502
[5] Norman, M. L.; Winkler, K.-H. A., (Proceedings of the NATO Advanced Workshop Held in Garching, August 2-13, 1982. Proceedings of the NATO Advanced Workshop Held in Garching, August 2-13, 1982, NATO ASI Series C, vol. 188 (1986)), 449-476
[6] Martí, J. M.; Ibáñez, J. M.; Miralles, J. A., Phys. Rev. D, 43, 3794 (1991)
[7] Marquina, A.; Martí, J. M.; Ibán̈ez, J. M.; Miralles, J. A.; Donat, R., Astron. Astrophys., 258, 566-571 (1992)
[8] F. Eulderink, (Ph.D. thesis), Rijksuniverteit te Leiden, Leiden, Netherlands, 1993; F. Eulderink, (Ph.D. thesis), Rijksuniverteit te Leiden, Leiden, Netherlands, 1993
[9] Eulderink, F.; Mellema, G., Astron. Astrophys. Suppl., 110, 587-623 (1995)
[10] Anile, A. M., Relativistic Fluids and Magneto-Fluids (1989), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 1106.76003
[11] Font, J. A.; Ibáñez, J. M.; Martí, J. M.; Marquina, A., Astron. Astrophys., 282, 304-314 (1994)
[12] Dolezal, A.; Wong, S. S.M., J. Comput. Phys., 120, 266-277 (1995) · Zbl 0840.76047
[13] Font, J. A., Living Rev. Relativ., 11, 7 (2008) · Zbl 1166.83003
[14] Antón, L.; Miralles, J. A.; Martí, J. M.; Ibáñez, J. M.; Aloy, M. A.; Mimica, P., Astrophys. J. Suppl. Ser., 188, 1-31 (2010)
[15] Boscheri, W.; Dumbser, M.; Balsara, D. S., Internat. J. Numer. Methods Fluids, 76, 737-778 (2014)
[16] Zanotti, O.; Dumbser, M., Comput. Phys. Comm., 188, 110-127 (2015) · Zbl 1344.76058
[17] Zanotti, O.; Fambri, F.; Dumbser, M., Mon. Not. R. Astron. Soc., 452, 3010-3029 (2015)
[18] del Zanna, L.; Bucciantini, N., Astron. Astrophys., 390, 1177-1186 (2002) · Zbl 1209.76022
[19] del Zanna, L.; Bucciantini, N.; Londrillo, P., Astron. Astrophys., 400, 397-413 (2003) · Zbl 1222.76122
[20] Qin, T.; Shu, C.-W.; Yang, Y., J. Comput. Phys., 315, 323-347 (2016) · Zbl 1349.83037
[21] Castro Díaz, M. J.; Fernández-Nieto, E. D., SIAM J. Sci. Comput., 34, A2173-A2196 (2012) · Zbl 1253.65167
[22] Castro, M. J.; Gallardo, J. M.; Marquina, A., J. Sci. Comput., 60, 363-389 (2014) · Zbl 1299.76181
[23] Castro, M. J.; Gallardo, J. M.; Marquina, A., Appl. Math. Comput., 272, 347-368 (2016) · Zbl 1410.76325
[24] Dumbser, M.; Toro, E. F., Commun. Comput. Phys., 10, 635-671 (2011) · Zbl 1373.76125
[25] Osher, S.; Solomon, F., Math. Comp., 38, 339-374 (1982) · Zbl 0483.65055
[26] Balsara, D., Astrophys. J. Suppl., 132, 83-101 (2001)
[27] Roe, P. L., J. Comput. Phys., 43, 357-372 (1981) · Zbl 0474.65066
[28] Cordier, F.; Degond, P.; Kumbaro, A., J. Sci. Comput., 58, 115-148 (2014) · Zbl 1302.76114
[29] Degond, P.; Peyrard, P. F.; Russo, G.; Villedieu, Ph., C. R. Acad. Sci. Paris Sér. I, 328, 479-483 (1999) · Zbl 0933.65101
[30] Torrilhon, M., SIAM J. Sci. Comput., 34, A2072-A2091 (2012) · Zbl 1253.65137
[31] Cargo, P.; Gallice, G., J. Comput. Phys., 136, 446-466 (1997) · Zbl 0919.76053
[32] Toro, E. F., (Riemann Solvers and Numerical Methods for Fluid Dynamics (2009), Springer) · Zbl 1227.76006
[33] Giacomazzo, B.; Rezzolla, L., J. Fluid Mech., 562, 223-259 (2006) · Zbl 1097.76073
[34] Marquina, A., SIAM J. Sci. Comput., 15, 892-915 (1994) · Zbl 0805.65088
[35] Shu, C.-W.; Osher, S., J. Comput. Phys., 77, 43971 (1998)
[36] Brio, M.; Wu, C. C., J. Comput. Phys., 75, 400-422 (1988) · Zbl 0637.76125
[37] Brackbill, J. U.; Barnes, J. C., J. Comput. Phys., 35, 426-430 (1980) · Zbl 0429.76079
[38] Balsara, D. S.; Spicer, D. S., J. Comput. Phys., 149, 270-292 (1999) · Zbl 0936.76051
[39] Orszag, S. A.; Tang, C. M., J. Fluid Mech., 90, 129-143 (1979)
[40] Beckwith, K.; Stone, J. M., Astrophys. J. Suppl., 193, 6 (2011)
[41] Komissarov, S. S., Mon. Not. R. Astron. Soc., 303, 343-366 (1999)
[42] Mignone, A.; Ugliano, M.; Bodo, G., Mon. Not. R. Astron. Soc., 393, 1141-1156 (2009)
[43] Radice, D.; Rezzolla, L., Astron. Astrophys., 547, A26 (2012)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.