Total-variation-diminishing implicit-explicit Runge-Kutta methods for the simulation of double-diffusive convection in astrophysics. (English) Zbl 1241.85010
Summary: We put forward the use of total-variation-diminishing (or more generally, strong stability preserving) implicit-explicit Runge-Kutta methods for the time integration of the equations of motion associated with the semiconvection problem in the simulation of stellar convection. The fully compressible Navier-Stokes equation, augmented by continuity and total energy equations, and an equation of state describing the relation between the thermodynamic quantities, is semi-discretized in space by essentially non-oscillatory schemes and dissipative finite difference methods. It is subsequently integrated in time by Runge-Kutta methods which are constructed such as to preserve the total variation diminishing (or strong stability) property satisfied by the spatial discretization coupled with the forward Euler method. We analyse the stability, accuracy and dissipativity of the time integrators and demonstrate that the most successful methods yield a substantial gain in computational efficiency as compared to classical explicit Runge-Kutta methods.
MSC:
| 85A30 | Hydrodynamic and hydromagnetic problems in astronomy and astrophysics |
| 85-08 | Computational methods for problems pertaining to astronomy and astrophysics |
| 76E20 | Stability and instability of geophysical and astrophysical flows |
Keywords:
hydrodynamics; stellar convection and pulsation; double-diffusive convection; numerical methods; total-variation-diminishing; strong stability preserving; TVD; SSPSoftware:
ANTARESReferences:
| [1] | Ascher, U.; Ruuth, S.; Spiteri, R., Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations, Appl. Numer. Math., 25, 151-167 (1997) · Zbl 0896.65061 |
| [2] | Ascher, U.; Ruuth, S.; Wetton, T., Implicit-explicit methods for time-dependent partial differential equations, SIAM J. Numer. Anal., 32, 3, 797-823 (1995) · Zbl 0841.65081 |
| [3] | Butcher, J., The Numerical Analysis of Ordinary Differential Equations (1987), John Wiley: John Wiley Chichester, UK · Zbl 0616.65072 |
| [4] | Canuto, V., Turbulence in astrophysical and geophysical flows, (Hillebrandt, W.; Kupka, F., Interdisciplinary Aspects of Turbulence. Interdisciplinary Aspects of Turbulence, Lecture Notes in Physics, vol. 756 (2009), Springer-Verlag), 107-160 · Zbl 1151.76003 |
| [5] | Canuto, V., Stellar mixing. II. Double diffusion processes, Astron. Astrophys., 528, A77 (2011) · Zbl 1260.85001 |
| [6] | Chandrasekhar, S., Hydrodynamic and Hydromagnetic Stability (1961), Clarendon Press: Clarendon Press Oxford · Zbl 0142.44103 |
| [7] | Cowling, T., The non-radial oscillations of polytropic stars, Mon. Not. Roy. Astron. Soc., 101, 367-375 (1941) · JFM 68.0607.01 |
| [8] | Donat, R.; Marquina, A., Capturing shock reflections: an improved flux formula, J. Comput. Phys., 125, 42-58 (1996) · Zbl 0847.76049 |
| [9] | Fehlberg, E., Klassische Runge-Kutta-Formeln vierter und niedrigerer Ordnung mit Schrittweiten-Kontrolle und ihre Anwendung auf Wärmeleitungsprobleme, Computing, 6, 61-71 (1970) · Zbl 0217.53001 |
| [10] | Ferracina, L.; Spijker, M., Strong stability of singly-diagonally-implicit Runge-Kutta methods, Appl. Numer. Math., 58, 1675-1686 (2008) · Zbl 1153.65080 |
| [11] | Ferziger, J.; Perić, M., Computational Methods for Fluid Dynamics (2002), Springer-Verlag: Springer-Verlag Berlin-Heidelberg-New York · Zbl 0998.76001 |
| [12] | Gottlieb, S.; Gottlieb, L.-A., Strong stability preserving properties of Runge-Kutta time discretization methods for linear constant coefficient operators, J. Sci. Comput., 18, 83-109 (2003) · Zbl 1030.65099 |
| [13] | Gottlieb, S.; Ketcheson, D.; Shu, C.-W., High order strong stability preserving time discretizations, J. Sci. Comput., 38, 251-289 (2009) · Zbl 1203.65135 |
| [14] | H. Grimm-Strele, Numerical solution of the generalised Poisson equation on parallel computers, Master’s Thesis, University of Vienna, March 2010. Available from: <http://othes.univie.ac.at/9200/>; H. Grimm-Strele, Numerical solution of the generalised Poisson equation on parallel computers, Master’s Thesis, University of Vienna, March 2010. Available from: <http://othes.univie.ac.at/9200/> |
| [15] | Hairer, E.; Nørsett, S.; Wanner, G., Solving Ordinary Differential Equations I (1987), Springer-Verlag: Springer-Verlag Berlin-Heidelberg-New York · Zbl 0638.65058 |
| [16] | Hairer, E.; Wanner, G., Solving Ordinary Differential Equations II (1991), Springer-Verlag: Springer-Verlag Berlin-Heidelberg-New York · Zbl 0729.65051 |
| [17] | N. Happenhofer, Simulation of low mach number fluids, Master’s Thesis, University of Vienna, June 2010. Available from: <http://othes.univie.ac.at/10551/>; N. Happenhofer, Simulation of low mach number fluids, Master’s Thesis, University of Vienna, June 2010. Available from: <http://othes.univie.ac.at/10551/> |
| [18] | Heun, K., Neue Methode zur approximativen Integration der Differentialgleichungen einer unabhängigen Veränderlichen, Zeitschrift f. Mathematik u. Physik, 45, 23-38 (1900) · JFM 31.0333.02 |
| [19] | I. Higueras, Radius and regions for arbitrary Γ;; I. Higueras, Radius and regions for arbitrary Γ; |
| [20] | Higueras, I., Representations of Runge-Kutta methods and strong stability preserving methods, SIAM J. Numer. Anal., 43, 924-948 (2005) · Zbl 1097.65078 |
| [21] | Higueras, I., Strong stability for additive Runge-Kutta methods, SIAM J. Numer. Anal., 44, 1735-1758 (2006) · Zbl 1126.65067 |
| [22] | Higueras, I., Characterizing strong stability preserving additive Runge-Kutta methods, J. Sci. Comput., 39, 115-128 (2009) · Zbl 1203.65109 |
| [23] | Hujeirat, A.; Rannacher, R., On the efficiency and robustness of implicit methods in computational astrophysics, New Astron. Rev., 45, 425-447 (2001) |
| [24] | Huppert, H.; Linden, P., On heating a stable salinity gradient from below, J. Fluid Mech., 95, 431-464 (1979) |
| [25] | Kennedy, C.; Carpenter, M., Additive Runge-Kutta schemes for convection-diffusion-reaction equations, Appl. Numer. Math., 44, 139-181 (2003) · Zbl 1013.65103 |
| [26] | Kippenhahn, R.; Weigert, A., Stellar Structure and Evolution (1994), Springer-Verlag |
| [27] | O. Koch, F. Kupka, B. Löw-Baselli, A. Mayrhofer, F. Zaussinger, SDIRK methods for the ANTARES code, ASC Report 32/2010, Inst. for Anal. and Sci. Comput., Vienna Univ. of Technology, 2010. Available from: <http://www.asc.tuwien.ac.at/preprint/2010/asc32x2010.pdf>; O. Koch, F. Kupka, B. Löw-Baselli, A. Mayrhofer, F. Zaussinger, SDIRK methods for the ANTARES code, ASC Report 32/2010, Inst. for Anal. and Sci. Comput., Vienna Univ. of Technology, 2010. Available from: <http://www.asc.tuwien.ac.at/preprint/2010/asc32x2010.pdf> |
| [28] | Kraaijevanger, J., Contractivity of Runge-Kutta methods, BIT, 31, 482-528 (1991) · Zbl 0763.65059 |
| [29] | Kupka, F.; Ballot, J.; Muthsam, H., Effects of resolution and Helium abundance in A star surface convection simulations, Commun. Asteroseismol., 160, 30-63 (2009) |
| [30] | Kwatra, N.; Su, J.; Grétarsson, J.; Fedkiw, R., A method for avoiding the acoustic time step restriction in compressible flow, J. Comput. Phys., 228, 4146-4161 (2009) · Zbl 1273.76356 |
| [31] | Liu, X.; Osher, S., Convex ENO high order multi-dimensional schemes without field by field decomposition or staggered grids, J. Comput. Phys., 142, 304-330 (1998) · Zbl 0941.65082 |
| [32] | Kang, X.-D. L.M.; Fedkiw, R. P., A boundary condition capturing method for multiphase incompressible flow, J. Sci. Comput., 15, 323-360 (2000) · Zbl 1049.76046 |
| [33] | Muthsam, H.; Göb, W.; Kupka, F.; Liebich, W., Interacting convection zones, New Astron., 4, 405-417 (1999) |
| [34] | Muthsam, H.; Göb, W.; Kupka, F.; Liebich, W.; Zöchling, J., A numerical study of compressible convection, Astron. Astrophys., 293, 127-141 (1995) |
| [35] | Muthsam, H.; Kupka, F.; Löw-Baselli, B.; Obertscheider, C.; Langer, M.; Lenz, P., ANTARES — A Numerical Tool for Astrophysical RESearch with applications to solar granulation, New Astron., 15, 460-475 (2010) |
| [36] | Pareschi, L.; Russo, G., Implicit-explicit Runge-Kutta schemes and application to hyperbolic systems with relaxation, J. Sci. Comput., 25, 129-155 (2005) · Zbl 1203.65111 |
| [37] | Ruuth, S.; Spiteri, R., High-order strong-stability-preserving Runge-Kutta methods with downwind-biased spatial discretizations, SIAM J. Numer. Anal., 42, 974-996 (2004) · Zbl 1089.65069 |
| [38] | Schwarzschild, M.; Härm, M., Evolution of very massive stars, Astrophys. J., 128, 348-360 (1958) |
| [39] | Shu, C.-W., Total-variation-diminishing time discretizations, SIAM J. Sci. Stat. Comput., 9, 1073-1084 (1988) · Zbl 0662.65081 |
| [40] | C.-W. Shu, Essentially non-oscillarory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, Tech. Rep. ICASE 97-65, Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, Hampton, VA, 1997.; C.-W. Shu, Essentially non-oscillarory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, Tech. Rep. ICASE 97-65, Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, Hampton, VA, 1997. |
| [41] | Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., 77, 439-471 (1988) · Zbl 0653.65072 |
| [42] | Spiteri, R.; Ruuth, S., A new class of optimal high-order strong-stability-preserving time discretization methods, SIAM J. Numer. Anal., 40, 469-491 (2002) · Zbl 1020.65064 |
| [43] | Spruit, H., The rate of mixing in semiconvective zones, Astron. Astrophys., 253, 131-138 (1992) · Zbl 0742.76078 |
| [44] | Spruit, H.; Nordlund, Å.; Title, A., Solar convection, Ann. Rev. Astron. Astrophys., 28, 263-301 (1990) |
| [45] | Strikwerda, J., Finite Difference Schemes and Partial Differential Equations (2004), SIAM: SIAM Philadelphia, PA · Zbl 1071.65118 |
| [46] | Turner, J., Multicomponent convection, Ann. Rev. Fluid Mech., 17, 11-44 (1985) |
| [47] | Weiss, A.; Hillebrandt, W.; Thomas, H.; Ritter, H., Cox and Giuli’s Principles of Stellar Structure, (Advances in Astronomy and Astrophysics (2004), Cambridge Scientific Publishers Ltd.: Cambridge Scientific Publishers Ltd. Cambridge, UK) |
| [48] | F. Zaussinger, Numerical simulation of double-diffusive convection, Ph.D. Thesis, University of Vienna, December 2010. Available from: <http://othes.univie.ac.at/13172/>; F. Zaussinger, Numerical simulation of double-diffusive convection, Ph.D. Thesis, University of Vienna, December 2010. Available from: <http://othes.univie.ac.at/13172/> |
| [49] | F. Zaussinger, H. Spruit, Semiconvection, Astron. Astrophys., submitted for publication. Available from: <arXiv:1012.5851v2>; F. Zaussinger, H. Spruit, Semiconvection, Astron. Astrophys., submitted for publication. Available from: <arXiv:1012.5851v2> |
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