On the correctness of numerical simulation of gravitational instability with the evolution of multiple gravitational collapses
Keywords:well-posedness, numerical methods, gravitational gas dynamics, instability, stability with respect to the input data, stochastics, Smoothed-Particle Hydrodynamics (SPH), circumstellar disc, gravitational collapse, gas clump
We study the correctness of numerical models based on the SPH-method for nonstationary problems of gravitational gas dynamics with the development of gravitational instability, including with the formation of multiple collapses of a gas in a circumstellar disk. The original differential initial boundary value problems are ill-posed because of their instability with respect to variations of input data. It is shown that the following regularization is performed in the SPH-based numerical method of solving such problems: (1) if the solution to the original unstable problem exists on the entire time axis, then the condition of the method’s stability with respect to small variations of input data is satisfied due to a bounded range of the variables; (2) if the solution of the original unstable problem exists only on a bounded time interval, as in the case of multiple collapses, then a stable numerical method can be developed on a class of functions bounded by a fixed constant chosen from physical considerations. On this class of bounded functions, the original problem becomes well-posed. A combination of the SPH method with a grid-based method of calculating the gravitational forces allows one to ensure the boundedness of numerical solutions. In order to clarify a meaning of the approximate numerical solutions obtained by computer simulations, it is necessary to use the integral functions weakly sensitive to the details of the numerical algorithm in use.
- A. C. Boley, T. Hayfield, L. Mayer, and R. H. Durisend, “Clumps in the Outer Disk by Disk Instability: Why They are Initially Gas Giants and the Legacy of Disruption,” Icarus 207 (2), 509-516 (2010).
- A. P. Boss, “Possible Rapid Gas Giant Planet Formation in the Solar Nebula and Other Protoplanetary Disks,” Astrophys. J. 536 (2), L101-L104 (2000).
- S. Nayakshin, S.-H. Cha, and J. C. Bridges, “The Tidal Downsizing Hypothesis for Planet Formation and the Composition of Solar System Comets,” Mon. Not. R. Astron. Soc. 416 (1), L50-L54 (2011).
- M. N. Machida, S.-I. Inutsuka, and T. Matsumoto, “Recurrent Planet Formation and Intermittent Protostellar Outflows Induced by Episodic Mass Accretion,” Astrophys. J. 729 (1), 1-17 (2011).
- F. Meru, “Triggered Fragmentation in Self-Gravitating Discs: Forming Fragments at Small Radii,” Mon. Not. R. Astron. Soc. 454 (3), 2529-2538 (2015).
- V. N. Snytnikov and O. P. Stoyanovskaya, “Clump Formation Due to the Gravitational Instability of a Multiphase Medium in a Massive Protoplanetary Disc,” Mon. Not. R. Astron. Soc. 428 (1), 2-12 (2013).
- E. I. Vorobyov and S. Basu, “Formation and Survivability of Giant Planets on Wide Orbits,” Astrophys. J. Lett. 714 (1), L133-L137 (2010).
- Z. Zhu, L. Hartmann, R. P. Nelson, and C. F. Gammie, “Challenges in Forming Planets by Gravitational Instability: Disk Irradiation and Clump Migration, Accretion, and Tidal Destruction,” Astrophys. J. 746 (1), 1-26 (2012).
- M. D. Young and C. J. Clarke, “Dependence of Fragmentation in Self-Gravitating Accretion Discs on Small-Scale Structure,” Mon. Not. R. Astron. Soc. 451 (4), 3987-3994 (2015).
- S. Z. Takahashi, Y. Tsukamoto, and S. Inutsuka, “A Revised Condition for Self-Gravitational Fragmentation of Protoplanetary Discs,” Mon. Not. R. Astron. Soc. 458 (4), 3597-3612 (2016).
- F. Meru and M. R. Bate, “Non-Convergence of the Critical Cooling Time-Scale for Fragmentation of Self-Gravitating Discs,” Mon. Not. R. Astron. Soc. 411 (1), L1-L5 (2011).
- A. A. Samarskii and Yu. P. Popov, Difference Methods for Solving the Gas Dynamics Problems (Nauka, Moscow, 1980) [in Russian].
- A. Lichtenberg and M. Lieberman, Regular and Chaotic Dynamics (Springer, New York, 1983; Mir, Moscow, 1984).
- G. M. Zaslavskii and R. Z. Sagdeev, Introduction to Nonlinear Physics. From the Pendulum to Turbulence and Chaos (Nauka, Moscow, 1988) [in Russian].
- O. P. Stoyanovskaya and V. N. Snytnikov, “Numerical Modeling of Formation of High Density Solitary Vortices in a Circumstellar Disk,” Vychisl. Metody Programm. 13, 377-383 (2012).
- M. R. Bate and A. Burkert, “Resolution Requirements for Smoothed Particle Hydrodynamics Calculations with Self-Gravity,” Mon. Not. R. Astron. Soc. 288 (4), 1060-1072 (1997).
- A. F. Nelson, “Numerical Requirements for Simulations of Self-Gravitating and Non-Self-Gravitating Discs,” Mon. Not. R. Astron. Soc. 373 (3), 1039-1073 (2006).
- J. K. Truelove, R. I. Klein, C. F. McKee, et al., “The Jeans Condition: A New Constraint on Spatial Resolution in Simulations of Isothermal Self-Gravitational Hydrodynamics,” Astrophys. J. 489 (2), L179-L183 (1997).
- E. I. Vorobyov, “Embedded Protostellar Disks Around (Sub-)Solar Protostars. I. Disk Structure and Evolution,” Astrophys. J. 2010. 723 (2), 1294-1307 (2010).
- K. Rohlfs, Lectures on Density Wave Theory (Springer, Berlin, 1977; Mir, Moscow, 1980).
- O. I. Bogoyavlenskii, Methods in the Qualitative Theory of Dynamical Systems in Astrophysics and Gas Dynamics (Nauka, Moscow, 1980; Springer, Berlin, 1985).
- O. P. Stoyanovskaya, N. V. Snytnikov, and V. N. Snytnikov, “An Algorithm for Solving Transient Problems of Gravitational Gas Dynamics: A Combination of the SPH Method with a Grid Method of Gravitational Potential Computation,” Vychisl. Metody Programm. 16, 52-60 (2015).
- J. J. Monaghan, “Smoothed Particle Hydrodynamics,” Annu. Rev. Astron. Astrophys. 30, 543-574 (1992).
- R. W. Hockney and J. W. Eastwood, Computer Simulation Using Particles (McGraw-Hill, New York, 1981).
- Yu. A. Berezin and V. A. Vshivkov, Particle-in-Cell Methods in Rarefied Plasma Dynamics (Nauka, Novosibirsk, 1980) [in Russian]
- V. L. Polyachenko and A. M. Fridman, Equilibrium and Stability of Gravitating Systems (Nauka, Moscow, 1976) [in Russian].
- O. P. Stoyanovskaya, “Numerical Simulation of Gravitational Instability Development and Clump Formation in Massive Circumstellar Disks Using Integral Characteristics for the Interpretation of Results,” Vychisl. Metody Programm. 17, 339-352 (2016).
- V. A. Vshivkov, S. A. Nikitin, and V. N. Snytnikov, “Studying Instability of Collisionless Systems on Stochastic Trajectories,” Pis’ma Zh. Eksp. Teor. Fiz. 78 (6), 810-815 (2003) [JETP Lett. 78 (6), 358-362 (2003)].
- D. V. Bisikalo, A. G. Zhilkin, and A. A. Boyarchuk, Gas Dynamics of Close Binary Stars (Fizmatlit, Moscow, 2013) [in Russian].
- V. S. Vladimirov, Equations of Mathematical Physics (Nauka, Moscow, 1971; Marcel Dekker, New York, 1971).
- S. I. Kabanikhin, Inverse and Ill-Posed Problems (Sibirsk. Nauchn. Izd., Novosibirsk, 2009) [in Russian].
- A. N. Tikhonov and V. Ya. Arsenin, Solution of Ill-Posed Problems (Nauka, Moscow, 1974; Wiley, New York, 1977).
- A. V. Goncharskii, A. M. Cherepashchuk, and A. G. Yagola, Ill-Posed Problems in Astrophysics (Nauka, Moscow, 1985) [in Russian].
- M. Schüssler and D. Schmitt, “Comments on Smoothed Particle Hydrodynamics,” Astron. Astrophys. 97 (2), 373-379 (1981).
- S. K. Godunov, Equations of Mathematical Physics (Nauka, Moscow, 1971) [in Russian].
- G. Korn and T. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968; Nauka, Moscow, 1973).