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A high order special relativistic hydrodynamic and magnetohydrodynamic code with space-time adaptive mesh refinement. (English) Zbl 1344.76058
Summary: We present a high order one-step ADER-WENO finite volume scheme with space-time adaptive mesh refinement (AMR) for the solution of the special relativistic hydrodynamic and magnetohydrodynamic equations. By adopting a local discontinuous Galerkin predictor method, a high order one-step time discretization is obtained, with no need for Runge-Kutta sub-steps. This turns out to be particularly advantageous in combination with space-time adaptive mesh refinement, which has been implemented following a “cell-by-cell” approach. As in existing second order AMR methods, also the present higher order AMR algorithm features time-accurate local time stepping (LTS), where grids on different spatial refinement levels are allowed to use different time steps.
We also compare two different Riemann solvers for the computation of the numerical fluxes at the cell interfaces. The new scheme has been validated over a sample of numerical test problems in one, two and three spatial dimensions, exploring its ability in resolving the propagation of relativistic hydrodynamical and magnetohydrodynamical waves in different physical regimes. The astrophysical relevance of the new code for the study of the Richtmyer-Meshkov instability is briefly discussed in view of future applications.

MSC:
76M12 Finite volume methods applied to problems in fluid mechanics
76W05 Magnetohydrodynamics and electrohydrodynamics
76Y05 Quantum hydrodynamics and relativistic hydrodynamics
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
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References:
[1] Martí, J. M.; Müller, E., Living Rev. Relativ., 6, 7, (2003)
[2] Font, J. A., Living Rev. Relativ., 6, 4, (2008)
[3] Dolezal, A., J. Comput. Phys., 120, 266-277, (1995)
[4] Del Zanna, L.; Bucciantini, N., Astronom. Astrophys., 390, 1177-1186, (2002)
[5] Del Zanna, L.; Bucciantini, N.; Londrillo, P., Astronom. Astrophys., 400, 397-413, (2003)
[6] Del Zanna, L.; Zanotti, O.; Bucciantini, N.; Londrillo, P., Astronom. Astrophys., 473, 11-30, (2007)
[7] Tchekhovskoy, A.; McKinney, J. C.; Narayan, R., Mon. Not. R. Astron. Soc., 379, 469-497, (2007)
[8] Bucciantini, N.; Del Zanna, L., Astronom. Astrophys., 528, A101, (2011)
[9] Radice, D.; Rezzolla, L., Phys. Rev. D, 84, 024010, (2011)
[10] Radice, D.; Rezzolla, L., Astronom. Astrophys., 547, A26, (2012)
[11] Dumbser, M.; Enaux, C.; Toro, E. F., J. Comput. Phys., 227, 3971-4001, (2008)
[12] Toro, E. F.; Titarev, V. A., Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 458, 271-281, (2002)
[13] Dumbser, M.; Balsara, D. S.; Toro, E. F.; Munz, C. D., J. Comput. Phys., 227, 8209-8253, (2008)
[14] Dumbser, M.; Zanotti, O., J. Comput. Phys., 228, 6991-7006, (2009)
[15] Choptuik, M. W., (Evans, C.; Finn, L.; Hobill, D., Frontiers in Numerical Relativity, (1989), Cambridge University Press Cambridge, England), 206-221
[16] Winkler, K. H.A.; Norman, M. L.; Mihalas, D., J. Quant. Spectrosc. Radiat. Transfer, 31, 473-478, (1984)
[17] Dönmez, O., Astrophys. Space Sci., 293, 323-354, (2004)
[18] Evans, E.; Iyer, S.; Schnetter, E.; Suen, W. M.; Tao, J.; Wolfmeyer, R.; Zhang, H. M., Phys. Rev. D, 71, 081301(R), (2005)
[19] van der Holst, B.; Keppens, R.; Meliani, Z., Comput. Phys. Comm., 179, 617-627, (2008)
[20] J. Tao, W.M. Suen, R. Wolfmeyer, H.M. Zhang, in: APS April Meeting Abstracts, p. K1012.
[21] Etienne, Z. B.; Liu, Y. T.; Shapiro, S. L., Phys. Rev. D, 82, 084031, (2010)
[22] De Colle, F.; Granot, J.; López-Cámara, D.; Ramirez-Ruiz, E., Astrophys. J., 746, 122, (2012)
[23] Liebling, S. L.; Lehner, L.; Neilsen, D.; Palenzuela, C., Phys. Rev. D, 81, 124023, (2010)
[24] Lehner, L.; Palenzuela, C.; Liebling, S. L.; Thompson, C.; Hanna, C., Phys. Rev. D, 86, 104035, (2012)
[25] East, W. E.; Pretorius, F.; Stephens, B. C., Phys. Rev. D, 85, 124010, (2012)
[26] Mignone, A.; Zanni, C.; Tzeferacos, P.; van Straalen, B.; Colella, P.; Bodo, G., Astrophys. J. Suppl. Ser., 198, 7, (2012)
[27] Berger, M. J.; Oliger, J., J. Comput. Phys., 53, 484-512, (1984)
[28] Berger, M. J., SIAM J. Sci. Stat. Comput., 7, 904-916, (1986)
[29] Berger, M. J.; Colella, P., J. Comput. Phys., 82, 64-84, (1989)
[30] Anderson, M.; Hirschmann, E.; Liebling, S. L.; Neilsen, D., Classical Quantum Gravity, 23, 6503-6524, (2006)
[31] Zhang, W.; MacFadyen, A., Astrophys. J. Suppl. Ser., 164, 255, (2006)
[32] MacNeice, P.; Olson, K. M.; Mobarry, C.; de Fainchtein, R.; Packer, C., Comput. Phys. Comm., 126, 330-354, (2000)
[33] Dumbser, M.; Zanotti, O.; Hidalgo, A.; Balsara, D. S., J. Comput. Phys., 248, 257-286, (2013)
[34] Dumbser, M.; Käser, M.; Toro, E. F., Geophys. J. Int., 171, 695-717, (2007)
[35] Taube, A.; Dumbser, M.; Munz, C. D.; Schneider, R., Int. J. Numer. Modelling, Electron. Netw. Devices Fields, 22, 77-103, (2009), Cited By (since 1996):9
[36] Lörcher, F.; Gassner, G.; Munz, C. D., J. Sci. Comput., 32, 175-199, (2007)
[37] Gassner, G.; Lörcher, F.; Munz, C. D., J. Sci. Comput., 34, 260-286, (2008)
[38] Anile, A. M., Relativistic fluids and magneto-fluids, (1990), Cambridge University Press · Zbl 0701.76003
[39] Martí, J. M.; Ibáñez, J. M.; Miralles, J. A., Phys. Rev. D, 43, 3794, (1991)
[40] Komissarov, S. S., Mon. Not. R. Astron. Soc., 303, 343-366, (1999)
[41] Balsara, D., Astrophys. J. Suppl. Ser., 132, 83-101, (2001)
[42] Dedner, A.; Kemm, F.; Kröner, D.; Munz, C. D.; Schnitzer, T.; Wesenberg, M., J. Comput. Phys., 175, 645-673, (2002)
[43] Palenzuela, C.; Lehner, L.; Reula, O.; Rezzolla, L., Mon. Not. R. Astron. Soc., 394, 1727-1740, (2009)
[44] Dionysopoulou, K.; Alic, D.; Palenzuela, C.; Rezzolla, L.; Giacomazzo, B., Phys. Rev. D, 88, 044020, (2013)
[45] Aloy, M. A.; Ibáñez, J. M.; Martí, J. M.; Müller, E., Astrophys. J. Suppl., 122, 151-166, (1999)
[46] Rezzolla, L.; Zanotti, O., Relativistic hydrodynamics, (2013), Oxford University Press Oxford, UK · Zbl 1297.76002
[47] Balsara, D. S.; Spicer, D. S., J. Comput. Phys., 149, 270-292, (1999)
[48] Antón, L.; Miralles, J. A.; Martí, J. M.; Ibáñez, J. M.; Aloy, M. A.; Mimica, P., Astrophys. J. Suppl., 188, 1-31, (2010)
[49] Baiotti, L., Numerical relativity simulations of non-vacuum spacetimes in three dimensions, (2004), SISSA, International School for advanced studies, (Ph.D. thesis)
[50] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S. R., J. Comput. Phys., 71, 231-303, (1987)
[51] Jiang, G. S.; Shu, C. W., J. Comput. Phys., 126, 202-228, (1996)
[52] Toro, E. F., Riemann solvers and numerical methods for fluid dynamics, (2009), Springer-Verlag · Zbl 1227.76006
[53] Clain, S.; Diot, S.; Loubère, R., J. Comput. Phys., 230, 4028-4050, (2011)
[54] Martí, J. M.; Müller, E., J. Fluid Mech., 258, 317-333, (1994)
[55] Rezzolla, L.; Zanotti, O., J. Fluid Mech., 449, 395, (2001)
[56] Giacomazzo, B.; Rezzolla, L., J. Fluid Mech., 562, 223-259, (2006)
[57] Harten, A.; Lax, P. D.; van Leer, B., SIAM Rev., 25, 35, (1983)
[58] Dumbser, M.; Toro, E., Commun. Comput. Phys., 10, 635-671, (2011)
[59] Solin, P., (Partial Differential Equations and the Finite Element Method, Pure and Applied Mathematics, (2006), Wiley-Interscience) · Zbl 1092.65080
[60] Liu, X. D.; Osher, S.; Chan, T., J. Comput. Phys., 115, 200-212, (1994)
[61] Toro, E. F.; Titarev, V. A., J. Comput. Phys., 202, 196-215, (2005)
[62] Dumbser, M.; Kaeser, M.; Titarev, V. A.; Toro, E. F., J. Comput. Phys., 226, 204-243, (2007)
[63] Zanotti, O.; Dumbser, M., Mon. Not. R. Astron. Soc., 418, 1004-1011, (2011)
[64] Khokhlov, A. M.; Oran, E. S.; Wheeler, J. C., Astrophys. J., 478, 678, (1997)
[65] Löhner, R., Comput. Methods Appl. Mech. Engrg., 61, 323-338, (1987)
[66] Baeza, A.; Martínez-Gavara, A.; Mulet, P., Appl. Numer. Math., 62, 278-296, (2012)
[67] Baeza, A.; Mulet, P., Internat. J. Numer. Methods Fluids, 52, 455-471, (2006)
[68] Anile, A. M.; Miller, J. C.; Motta, S., Phys. Fluids, 26, 1450-1460, (1983)
[69] Mignone, A.; Bodo, G., Mon. Not. R. Astron. Soc., 364, 126-136, (2005)
[70] Rezzolla, L.; Zanotti, O., Phys. Rev. Lett., 89, 114501, (2002)
[71] Aloy, M. A.; Rezzolla, L., Astrophys. J., 640, L115-L118, (2006)
[72] Rezzolla, L.; Zanotti, O.; Pons, J. A., J. Fluid Mech., 479, 199, (2003)
[73] Lucas-Serrano, A.; Font, J. A.; Ibáñez, J. M.; Marti, J. M., Astronom. Astrophys., 428, 703-715, (2004)
[74] Mignone, A.; Ugliano, M.; Bodo, G., Mon. Not. R. Astron. Soc., 393, 1141-1156, (2009)
[75] Beckwith, K.; Stone, J. M., Astrophys. J. Suppl. Ser., 193, 6, (2011)
[76] Richtmyer, R. D., Comm. Pure Appl. Math., 13, 297-319, (1960)
[77] Meshkov, E., Fluid Dyn., 43, 101-104, (1968)
[78] Taub, A. H., Phys. Rev., 74, 328-334, (1948)
[79] Brouillette, M., Annu. Rev. Fluid Mech., 34, 445-468, (2002)
[80] F. Mohseni, M. Mendoza, S. Succi, H.J. Herrmann, 2013. ArXiv e-prints.
[81] Sod, G. A., J. Comput. Phys., 27, 1-31, (1978)
[82] Komissarov, S. S., Phys. Lett. A, 232, 435-442, (1997)
[83] Orszag, S. A.; Tang, C. M., J. Fluid Mech., 90, 129, (1979)
[84] Dahlburg, R. B.; Picone, J. M., Phys. Fluids B, 1, 2153-2171, (1989)
[85] Jiang, G.; Wu, C., J. Comput. Phys., 150, 561-594, (1999)
[86] Inoue, T., Astrophys. J., 760, 43, (2012)
[87] Matsumoto, J.; Masada, Y., Astrophys. J. Lett., 772, L1, (2013)
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