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An algorithm for computing the 2D structure of fast rotating stars. (English) Zbl 1349.85004
Summary: Stars may be understood as self-gravitating masses of a compressible fluid whose radiative cooling is compensated by nuclear reactions or gravitational contraction. The understanding of their time evolution requires the use of detailed models that account for a complex microphysics including that of opacities, equation of state and nuclear reactions. The present stellar models are essentially one-dimensional, namely spherically symmetric. However, the interpretation of recent data like the surface abundances of elements or the distribution of internal rotation have reached the limits of validity of one-dimensional models because of their very simplified representation of large-scale fluid flows. In this article, we describe the ESTER code, which is the first code able to compute in a consistent way a two-dimensional model of a fast rotating star including its large-scale flows.compared to classical 1D stellar evolution codes, many numerical innovations have been introduced to deal with this complex problem. First, the spectral discretization based on spherical harmonics and Chebyshev polynomials is used to represent the 2D axisymmetric fields. A nonlinear mapping maps the spheroidal star and allows a smooth spectral representation of the fields. The properties of Picard and Newton iterations for solving the nonlinear partial differential equations of the problem are discussed. It turns out that the Picard scheme is efficient on the computation of the simple polytropic stars, but Newton algorithm is unsurpassed when stellar models include complex microphysics. Finally, we discuss the numerical efficiency of our solver of Newton iterations. This linear solver combines the iterative Conjugate Gradient Squared algorithm together with an LU-factorization serving as a preconditioner of the Jacobian matrix.

85A15 Galactic and stellar structure
76N15 Gas dynamics, general
76U05 General theory of rotating fluids
76W05 Magnetohydrodynamics and electrohydrodynamics
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[1] Aksenov, A. G.; Blinnikov, S. I., A Newton iteration method for obtaining equilibria of rapidly rotating stars, Astron. Astrophys., 290, 674-681, (Oct. 1994)
[2] Angulo, C.; Arnould, M.; Rayet, M.; Descouvemont, P.; Baye, D.; Leclercq-Willain, C.; Coc, A.; Barhoumi, S.; Aguer, P.; Rolfs, C.; Kunz, R.; Hammer, J. W.; Mayer, A.; Paradellis, T.; Kossionides, S.; Chronidou, C.; Spyrou, K.; degl’Innocenti, S.; Fiorentini, G.; Ricci, B.; Zavatarelli, S.; Providencia, C.; Wolters, H.; Soares, J.; Grama, C.; Rahighi, J.; Shotter, A.; Lamehi Rachti, M., A compilation of charged-particle induced thermonuclear reaction rates, Nucl. Phys. A, 656, 3-183, (Aug. 1999)
[3] Bonazzola, S.; Gourgoulhon, E.; Marck, J.-A., Numerical approach for high precision 3D relativistic star models, Phys. Rev. D, 58, (1998)
[4] Brott, I.; Evans, C. J.; Hunter, I.; de Koter, A.; Langer, N.; Dufton, P. L.; Cantiello, M.; Trundle, C.; Lennon, D. J.; de Mink, S. E.; Yoon, S.-C.; Anders, P., Rotating massive main-sequence stars, II: simulating a population of LMC early B-type stars as a test of rotational mixing, Astron. Astrophys., 530, (Jun. 2011)
[5] Busse, F., Do Eddington-sweet circulations exist?, Geophys. Astrophys. Fluid Dyn., 17, 215, (1981) · Zbl 0471.76111
[6] Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A., Spectral methods evolution to complex geometries and applications to fluid dynamics, (2007), Springer Verlag · Zbl 1121.76001
[7] Clement, M. J., On the solution of Poisson’s equation for rapidly rotating stars, Astrophys. J., 194, 709-714, (Dec. 1974)
[8] Clement, M. J., On the solution of the equilibrium equations for rapidly rotating stars, Astrophys. J., 222, 967-975, (Jun. 1978)
[9] Clement, M. J., On the equilibrium and secular instability of rapidly rotating stars, Astrophys. J., 230, 230-242, (May 1979)
[10] Clement, M. J., Differential rotation and the convective core mass of upper main-sequence stars, Astrophys. J., 420, 797-802, (Jan. 1994)
[11] Deupree, R. G., Structure of uniformly rotating stars, Astrophys. J., 735, 69, (Jul. 2011)
[12] Deupree, R. G.; Castañeda, D.; Peña, F.; Short, C. I., Matching the spectral energy distribution and P-mode oscillation frequencies of the rapidly rotating delta scuti star α ophiuchi with a two-dimensional rotating stellar model, Astrophys. J., 753, 20, (Jul. 2012)
[13] Domiciano de Souza, A.; Kervella, P.; Jankov, S.; Abe, L.; Vakili, F.; di Folco, E.; Paresce, F., The spinning-top be star achernar from VLTI-VINCI, Astron. Astrophys., 407, L47-L50, (Aug. 2003)
[14] Einset, E. O.; Jensen, K. F., A finite element solution of three-dimensional mixed convection gas flows in horizontal channels using preconditioned iterative matrix methods, Int. J. Numer. Methods Fluids, 14, 817-841, (1992) · Zbl 0825.76443
[15] Eriguchi, Y.; Müller, E., A general computational method for obtaining equilibria of self-gravitating and rotating gases, Astron. Astrophys., 146, 260-268, (May 1985)
[16] Eriguchi, Y.; Müller, E., Structure of rapidly rotating axisymmetric stars, I: a numerical method for stellar structure and meridional circulation, Astron. Astrophys., 248, 435-447, (Aug. 1991)
[17] Espinosa Lara, F.; Rieutord, M., The dynamics of a fully radiative rapidly rotating star enclosed within a spherical box, Astron. Astrophys., 470, 1013-1022, (2007) · Zbl 1125.85002
[18] Espinosa Lara, F.; Rieutord, M., Self-consistent 2D models of fast rotating early-type stars, Astron. Astrophys., 552, (2013)
[19] Gourgoulhon, E.; Grandclément, P.; Taniguchi, K.; Marck, J.-A.; Bonazzola, S., Quasiequilibrium sequences of synchronized and irrotational binary neutron stars in general relativity: method and tests, Phys. Rev. D, 63, 6, (2001)
[20] Greenspan, H. P., The theory of rotating fluids, (1968), Cambridge University Press · Zbl 0182.28103
[21] Hansen, C.; Kawaler, S., Stellar interiors: physical principles, structure and evolution, (1994), Springer
[22] Houdek, G.; Rogl, J., On the accuracy of opacity interpolation schemes, Bull. Astron. Soc. India, 24, 317, (Jun. 1996)
[23] Hui-Bon-Hoa, A., The Toulouse Geneva evolution code (TGEC), Astrophys. Space Sci., 316, 55-60, (Aug. 2008)
[24] Jackson, S.; MacGregor, K. B.; Skumanich, A., Models for the rapidly rotating be star achernar, Astrophys. J., 606, 1196-1199, (May 2004)
[25] Jackson, S.; MacGregor, K. B.; Skumanich, A., On the use of the self-consistent-field method in the construction of models for rapidly rotating main-sequence stars, Astrophys. J. Suppl. Ser., 156, 245-264, (Feb. 2005)
[26] James, R. A., The structure and stability of rotating gas masses, Astrophys. J., 140, 552, (Aug. 1964)
[27] Kippenhahn, R.; Weigert, A., Stellar structure and evolution, (1990), Springer · Zbl 1254.85001
[28] Kippenhahn, R.; Weigert, A.; Weiss, A., Stellar structure and evolution, (2012), Springer · Zbl 1253.85001
[29] Knoll, D. A.; Keyes, D. E., Jacobian-free Newton-Krylov methods: a survey of approaches and applications, J. Comput. Phys., 193, 357-397, (Jan. 2004)
[30] Lemieux, J.-F.; Price, S. F.; Evans, K. J.; Knoll, D.; Salinger, A. G.; Holland, D. M.; Payne, A. J., Implementation of the Jacobian-free Newton-Krylov method for solving the first-order ice sheet momentum balance, J. Comput. Phys., 230, 6531-6545, (Jul. 2011)
[31] MacGregor, K. B.; Jackson, S.; Skumanich, A.; Metcalfe, T. S., On the structure and properties of differentially rotating, main-sequence stars in the 1-2 \(\operatorname{M}_{s o l a r}\) range, Astrophys. J., 663, 560-572, (Jul. 2007)
[32] Maeder, A., Physics, formation and evolution of rotating stars, (2009), Springer
[33] Monnier, J. D.; Zhao, M.; Pedretti, E.; Thureau, N.; Ireland, M.; Muirhead, P.; Berger, J.-P.; Millan-Gabet, R.; Van Belle, G.; ten Brummelaar, T.; McAlister, H.; Ridgway, S.; Turner, N.; Sturmann, L.; Sturmann, J.; Berger, D., Imaging the surface of altair, Science, 317, 342-345, (Jul. 2007)
[34] Morel, P., CESAM: a code for stellar evolution calculations, Astron. Astrophys. Suppl. Ser., 124, 597-614, (1997)
[35] Morel, P.; Lebreton, Y., CESAM: a free code for stellar evolution calculations, Astron. Astrophys. Suppl. Ser., 316, 61-73, (Aug. 2008)
[36] Oliver, H. J.; Reiman, A. H.; Monticello, D. A., Solving the 3D MHD equilibrium equations in toroidal geometry by Newton’s method, J. Comput. Phys., 211, 99-128, (Jan. 2006)
[37] Ostriker, J. P.; Mark, J. W.-K., Rapidly rotating stars. I. the self-consistent-field method, Astrophys. J., 151, 1075-1088, (Mar. 1968)
[38] Paniconi, C.; Putti, M., A comparison of Picard and Newton iteration in the numerical solution of multidimensional variably saturated flow problems, Water Resour. Res., 30, 3357-3374, (Dec. 1994)
[39] Paxton, B.; Bildsten, L.; Dotter, A.; Herwig, F.; Lesaffre, P.; Timmes, F., Modules for experiments in stellar astrophysics (MESA), Astrophys. J. Suppl. Ser., 192, 3, (Jan. 2011)
[40] Rieutord, M., The dynamics of the radiative envelope of rapidly rotating stars, I: a spherical Boussinesq model, Astron. Astrophys., 451, 1025-1036, (2006) · Zbl 1096.85009
[41] Rieutord, M., Modeling rapidly rotating stars, (Casoli, F.; etal., SF2A Proceeding, (2006)) · Zbl 1096.85009
[42] Rieutord, M., On the dynamics of radiative zones in rotating star, (Rieutord, M.; Dubrulle, B., Stellar Fluid Dynamics and Numerical Simulations: From the Sun to Neutron Stars, EAS, vol. 21, (2006)), 275-295
[43] Rieutord, M.; Espinosa Lara, F., On the dynamics of a radiative rapidly rotating star, Commun. Asteroseismol., 158, 99-103, (2009)
[44] Rieutord, M.; Espinosa Lara, F., Ab initio modelling of steady rotating stars, (Goupil, M.; Belkacem, K.; Neiner, C.; Lignières, F.; Green, J. J., Seismology for Studies of Stellar Rotation and Convection, Lecture Notes in Physics, vol. 865, (2013), Springer Verlag Berlin), 49-73
[45] Rogers, F. J.; Swenson, F. J.; Iglesias, C. A., OPAL equation-of-state tables for astrophysical applications, Astrophys. J., 456, 902, (Jan. 1996)
[46] Roxburgh, I. W., 2-dimensional models of rapidly rotating stars, I: uniformly rotating zero age main sequence stars, Astron. Astrophys., 428, 171-179, (Dec. 2004)
[47] Roxburgh, I. W., 2-dimensional models of rapidly rotating stars, II: hydrostatic and acoustic models with \(\operatorname{\Omega} = \operatorname{\Omega}(r, \theta)\), Astron. Astrophys., 454, 883-888, (Aug. 2006)
[48] Sonneveld, P., CGS: a fast Lanczos-type solver for nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 10, 36-52, (1989) · Zbl 0666.65029
[49] Xu, Y.; Takahashi, K.; Goriely, S.; Arnould, M.; Ohta, M.; Utsunomiya, H., NACRE II: an update of the NACRE compilation of charged-particle-induced thermonuclear reaction rates for nuclei with mass number \(A < 16\), Nucl. Phys. A, 918, 61-169, (Nov. 2013)
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