Integral-Consistent Numerical Technique for Self-gravitating Medium Model

  • Yu. S. Sharova
  • Yu. A. Poveshchenko
  • V. A. Gasilov
  • N. S. Smirnova
  • V. O. PodrygaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11386)


The supercompression of matter caused by gravitational coupling or self-gravitational forces leads to density growth by several orders in magnitude. Keeping in mind the importance of self-gravitation in astrophysical processes like supernovae star evolution we consider it reasonable to develop a numerical technique based on the consistent approximation to the terms describing gravitational energy transfer into the kinetic energy of a matter in the star along its life cycle. The so-called completely conservative gas-dynamics difference schemes including the gravitation effects are the proper numerical technique able to simulate correctly the problems concerning gravitational coupling effects. The accounting for gravitational forces in the construction of completely conservative difference schemes is a significant complication. In the paper, we propose an integrally-consistent difference scheme that utilizes the method of support difference operators thus providing a possibility to conform the balance between kinetic and gravitational energy increments or losses. According to this method, we use the result of the total gravitational energy varying and construct the symmetrized strain rate tensor as the base operator. The result of varying the gravitational energy of the system is a discrete convolution of the Newton gravitational tensor in the difference media under study, which exhaustively answers all the gravitational processes unfolding against the background of the hydrodynamic motion of matter. The symmetrized strain tensor governs the kinematic motion in a considered system. The conjugate operator related to the convolution of these tensors automatically gives the approximation to the gravitational forces acting in the interior of the balance volume of the difference model built via the support operator approach.


Self-gravitation Mathematical modeling 



The work was funded by Russian Foundation for Basic Research, projects no. 18-07-00841-a, 16-29-15081-ofi_m, 16-29-15095-ofi_m.

Supported by the Academic Excellence Project of the NRNU MEPhI under contract with the Ministry of Education and Science of the Russian Federation No. 02.A03.21.0005.


  1. 1.
    Samarskii, A.A., Popov, Yu.P.: Difference methods for solving problems of gas dynamics. Nauka, Moscow (1992)Google Scholar
  2. 2.
    Denisov, A.A., Koldoba, A.V., Poveshchenko, Yu.A., Popov, Yu.P., Chechetkin, V.M.: The role of rotation in the thermonuclear model of supernova explosion. Preprints of the Keldysh Institute of Applied Mathematics, no. 99 (1986)Google Scholar
  3. 3.
    Koldoba, A.V., Kuznetsov, O.A., Poveshchenko, Yu.A., Popov, Yu.P., Samarskii, A.A.: Completely conservative difference schemes for the equations of continuum mechanics in quasi-Lagrangian variables in the presence of gravitational and magnetohydrodynamic processes. Preprints of the Keldysh Institute of Applied Mathematics, no. 55 (1985)Google Scholar
  4. 4.
    Kochin, N.E.: Vector calculus and initiate of tensor calculus. Lenand, Moscow (2017)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Yu. S. Sharova
    • 1
  • Yu. A. Poveshchenko
    • 1
    • 2
  • V. A. Gasilov
    • 1
    • 2
  • N. S. Smirnova
    • 3
  • V. O. Podryga
    • 1
    • 4
    Email author
  1. 1.Keldysh Institute of Applied Mathematics of RASMoscowRussia
  2. 2.National Research Nuclear University MEPhIMoscowRussia
  3. 3.Laboratoire d’Annecy-le-Vieux de Physique TheoriqueAnnecyFrance
  4. 4.Moscow Automobile and Road Construction State Technical UniversityMoscowRussia

Personalised recommendations