DOI: https://doi.org/10.26089/NumMet.v16r106

An algorithm for solving transient problems of gravitational gas dynamics: a combination of the SPH method with a grid method of gravitational potential computation

Authors

  • O.P. Stoyanovskaya
  • N.V. Snytnikov
  • V.N. Snytnikov

Keywords:

self-gravitating circumstellar disk
structure formation
solitary clumps
Smoothed-Particle Hydrodynamics (SPH)
Poisson’s equation
gravitational gas dynamics

Abstract

A new numerical algorithm to solve the unsteady equations of gravitational gas dynamics in the thin disk approximation is proposed. This algorithm is based on a combination of the meshless SPH (Smoothed Particle Hydrodynamics) method for gas dynamics and the convolution method for solving Poisson’s equation on a Cartesian grid. This convolution method is of high performance due to the fact that the grid potential function is computed and stored only in the plane of the disk. The efficiency of the algorithm is demonstrated by numerical experiments on the formation of structures in a circumstellar disk. We compare the results obtained by using the grid method for solving Poisson’s equation in Cartesian and cylindrical geometry and show that in both these cases it is possible to reproduce the solutions with axial symmetry and to illustrate the formation of solitary regions of enhanced density.


Published

2015-02-12

Issue

Section

Section 1. Numerical methods and applications

Author Biographies

O.P. Stoyanovskaya

N.V. Snytnikov

V.N. Snytnikov


References

  1. J. P. Ruge, S. Wolf, A. L. Uribe, and H. H. Klahr, “Tracing Planets in Circumstellar Discs. Observability of Large-Scale Disc Structures with ALMA,” Eur. Phys. J. Web Conf. 46 (2013).
    doi 10.1051/epjconf/20134602003
  2. V. A. Vshivkov and A. V. Snytnikov, “Development of an Efficient Parallel Poisson Equation Solver for the Simulation of Protoplanetary Disk Evolution,” Vychisl. Metody Programm. 10, 116-122 (2009).
  3. O. P. Stoyanovskaya and V. N. Snytnikov, “Features of SPH Gas Dynamics for Modeling of Nonlinear Gravitational Waves in Multiphase Medium,” Mat. Model. 22 (5), 29-44 (2010).
  4. V. A. Vshivkov and A. V. Snytnikov, “Numerical Modeling of Formation of High Density Solitary Vortices in a Circumstellar Disk,” Vychisl. Metody Programm. 13, 377-383 (2012).
  5. R. W. Hockney and J. W. Eastwood, Computer Simulation Using Particles (McGraw-Hill, New York, 1981).
  6. A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis (Nauka, Moscow, 1989; Dover Publ., Mineola, 1999).
  7. J. W. Eastwood and D. R. K. Brownrigg, “Remarks on the Solution of Poisson’s Equation for Isolated Systems,” J. Comput. Phys. 32 (1), 24-38 (1979).
  8. M. Frigo and S. G. Johnson, “The Design and Implementation of FFTW3,” Proc. IEEE 93 (2), 216-231 (2005).
  9. O. Ayala and L.-P. Wang, “Parallel Implementation and Scalability Analysis of 3D Fast Fourier Transform Using 2D Domain Decomposition,” Parallel Comput. 39 (1), 58-77 (2013).
  10. P. Barge and L. Jorda, “Instabilities and Structures in Proto-Planetary Disks,” Eur. Phys. J. Web Conf. 46 (2013).
    doi 10.1051/epjconf/20134600001
  11. W. K. M. Rice, G. Lodato, J. E. Pringle, et al., “Planetesimal Formation via Fragmentation in Self-Gravitating Protoplanetary Discs,” Mon. Not. R. Astron. Soc. 372, 9-13 (2006).
  12. A. C. Boley, T. Hayfield, L. Mayer, and R. H. Durisen, “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).
  13. E. I. Vorobyov and S. Basu, “Formation and Survivability of Giant Planets on Wide Orbits,” Astrophys. J. Lett. 714 (1), 133-137 (2010).
  14. F. Meru and M. R. Bate, “On the Convergence of the Critical Cooling Time-Scale for the Fragmentation of Self-Gravitating Discs,” Mon. Not. R. Astron. Soc. 427 (3), 2022-2046 (2012).
  15. 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).
  16. S. Nayakshin, R. Helled, and A. C. Boley, “Core-Assisted Gas Capture Instability: a New Mode of Giant Planet Formation by Gravitationally Unstable Discs.’’ Mon. Not. R. Astron. Soc. 440 (4), 3797-3808 (2014).
  17. G. Lodato and C. J. Clarke, “Resolution Requirements for Smoothed Particle Hydrodynamics Simulations of Self-Gravitating Accretion Discs,” Mon. Not. R. Astron. Soc. 413 (4), 2735-2740 (2011).