Skip to main content
SpringerLink
Log in

Modified method of splitting with respect to physical processes for solving radiation gas dynamics equations

  • Published:
Computational Mathematics and Mathematical Physics Aims and scope Submit manuscript

Abstract

An approach based on a modified splitting method is proposed for solving the radiation gas dynamics equations in the multigroup kinetic approximation. The idea of the approach is that the original system of equations is split using the thermal radiation transfer equation rather than the energy equation. As a result, analytical methods can be used to solve integrodifferential equations and problems can be computed in the multigroup kinetic approximation without iteration with respect to the collision integral or matrix inversion. Moreover, the approach can naturally be extended to multidimensional problems. A high-order accurate difference scheme is constructed using an approximate Godunov solver for the Riemann problem in two-temperature gas dynamics.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ya. B. Zel’dovich and Yu. P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena (Nauka, Moscow, 1966; Academic, New York, 1966, 1967), 2 Vols.

    Google Scholar 

  2. N. N. Yanenko, The Method of Fractional Steps: The Solution of Problems of Mathematical Physics in Several Variables (Nauka, Novosibirsk, 1967; Springer-Verlag, Berlin, 1971).

    MATH  Google Scholar 

  3. G. I. Marchuk, Splitting Methods (Nauka, Moscow, 1988) [in Russian].

    MATH  Google Scholar 

  4. B. N. Chetverushkin, Mathematical Simulation Radiating Gas Dynamics (Nauka, Moscow, 1985) [in Russian].

    MATH  Google Scholar 

  5. Shih-I Pai, Radiation Gas Dynamics (Springer-Verlag, Wien, 1966; Mir, Moscow, 1968).

    Book  MATH  Google Scholar 

  6. O. M. Belotserkovskii and Yu. M. Davydov, Large-Particle Method in Gas Dynamics (Fizmatlit, Moscow, 1982) [in Russian].

    Google Scholar 

  7. B. G. Carlson and G. I. Bell, “Solution of the transport equation by the Sn method,” in Proceeding of the Second Unified Nations International Conference on the Peaceful Uses of Atomic Energy, P/2386 (Unified Nations, 1958), Vol. 16, pp. 535–549.

    Google Scholar 

  8. B. G. Carlson and G. I. Bell, “Numerical solution of neutron transport problems,” in Proceedings of Symposia in Applied Mathematics, Vol. 11: Nuclear Reactor Theory (Am. Math. Soc., Providence, RI, 1961; Gosatomizdat, Moscow, 1963).

    Google Scholar 

  9. L. P. Bass, A. M. Voloshchenko, and T. A. Germogenova, Discrete Ordinate Methods for Radiative Transfer Problems (Inst. Prikl. Mat. im. M.V. Keldysha Ross. Akad. Nauk, Moscow, 1986) [in Russian].

    Google Scholar 

  10. V. S. Vladimirov, “Numerical solution of the equation for the sphere,” in Numerical Mathematics (Vychisl. Tsentr Akad. Nauk SSSR, Moscow, 1958), pp. 3–33 [in Russian].

    Google Scholar 

  11. A. V. Nikiforova, V. A. Tarasov, and V. E. Troshchiev, “Solution of the kinetic equations by the divergent method of characteristics,” USSR Comput. Math. Math. Phys. 12 (4), 251–260 (1972).

    Article  Google Scholar 

  12. P. C. R. Feautrier, “Sur la resolution numerique de l’equation de transfert,” Acad. Sci. Paris 258, 3189–3191 (1964).

    Google Scholar 

  13. G. Rybicki, “A modified Feautrier method,” J. Quant. Spectrosc. Radiat. Transfer 11, 589–596 (1971).

    Article  Google Scholar 

  14. V. Ya. Gol’din, “A quasi-diffusion method of solving the kinetic equation,” USSR Comput. Math. Math. Phys. 4 (6), 136–149 (1964).

    Article  MathSciNet  MATH  Google Scholar 

  15. D. Yu. Anistratov, E. N. Aristova, and V. Ya. Gol’din, “Nonlinear method for solving problems of radiative transfer in media,” Mat. Model. 8 (12), 3–28 (1996).

    MathSciNet  MATH  Google Scholar 

  16. A. I. Zuev, “Application of the Newton–Kantorovich method for solving the problem of propagation of nonequilibrium radiation,” USSR Comput. Math. Math. Phys. 13 (3), 338–346 (1973).

    Article  Google Scholar 

  17. V. Yu. Gusev, M. Yu. Kozmanov, and E. B. Rachilov, “A method of solving implicit difference equations approximating systems of radiation transport and diffusion equations,” USSR Comput. Math. Math. Phys. 24 (12), 156–161 (1984).

    Article  MATH  Google Scholar 

  18. L. P. Fedotova and R. M. Shagaliev, “Finite-difference KM method for two-dimensional unsteady transport processes in the multigroup transport approximation,” Mat. Model. 3 (6), 29–41 (1991).

    MathSciNet  MATH  Google Scholar 

  19. A. D. Gadzhiev, E. M. Romanova, V. N. Seleznev, and A. A. Shestakov, “TOM4-KD method for mathematical simulation of two-dimensional radiative transfer equations in the multigroup quasi-diffusion approximation,” Vopr. At. Nauki Tekh., Ser. Mat. Model. Fiz. Processov, No. 4, 48–59 (2001).

    Google Scholar 

  20. N. G. Karlykhanov, “Construction of optimal multidiagonal methods for solving radiation-transfer problems,” Comput. Math. Math. Phys. 37 (4), 482–486 (1997).

    MathSciNet  MATH  Google Scholar 

  21. G. V. Dolgoleva, “Numerical solution of the system of equations describing radiative transfer and the interaction of radiation with matter,” Vopr. At. Nauki Tekh., Ser. Mat. Model. Fiz. Processov, No. 1, 58–60 (1991).

    Google Scholar 

  22. A. A. Samarskii, Introduction to the Theory of Difference Schemes (Nauka, Moscow, 1971) [in Russian].

    Google Scholar 

  23. A. A. Samarskii and P. N. Vabishchevich, Additive Schemes for Problems in Mathematical Physics (Nauka, Moscow, 2001) [in Russian].

    MATH  Google Scholar 

  24. V. V. Smelov, Lectures on the Theory of Neutron Transport (Atomizdat, Moscow, 1978) [in Russian].

    Google Scholar 

  25. D. Mihalas, Stellar Atmospheres (Freeman, San Francisco, 1978; Mir, Moscow, 1982).

    Google Scholar 

  26. E. V. Groshev, “Application of Rybicki’s method to the iterative solution of radiative transfer equations using boundary conditions,” Vopr. At. Nauki Tekh., Ser. Mat. Model. Fiz. Processov, No. 1, 39–47 (2010).

    Google Scholar 

  27. A. D. Gadzhiev, V. N. Seleznev, and A. A. Shestakov, “DSn-method with artificial dissipation and dmt method of iteration acceleration for the numerical solution of two-dimensional heat transfer equations in kinetic approximation,” Vopr. At. Nauki Tekh., Ser. Mat. Model. Fiz. Processov, No. 4, 33–46 (2003).

    Google Scholar 

  28. R. E. Alcouff, “A stable diffusion synthetic acceleration method for neutron transport iterations,” Trans. Am. Nucl. Soc. 23, 203 (1976).

    Google Scholar 

  29. R. E. Alcouff, D. R. McCoy, and E. W. Larsen, “Finite difference effects in the synthetic acceleration method,” Trans. Am. Nucl. Soc. 39, 462 (1981).

    Google Scholar 

  30. T. A. Sushkevich, Mathematical Models of Radiative Transfer (Binom, Moscow, 2006) [in Russian].

    Google Scholar 

  31. N. Ya. Moiseev, “Explicit-implicit difference scheme for the joint solution of the radiative transfer and energy equations by the splitting method,” Comput. Math. Math. Phys. 53 (3), 320–335 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  32. N. Ya. Moiseev and V. M. Shmakov, “Modified splitting method for solving the nonstationary kinetic particle transport equation,” Comput. Math. Math. Phys. 56 (8), 1464–1473 (2016).

    Article  MathSciNet  Google Scholar 

  33. G. I. Marchuk and N. N. Yanenko, “Solution of a multidimensional kinetic equation by the splitting method,” Dokl. Akad. Nauk SSSR 157 (6), 1291–1292 (1964).

    MathSciNet  MATH  Google Scholar 

  34. S. K. Godunov, “Difference method for computing discontinuous solutions of fluid dynamics equations,” Mat. Sb. 47 (3), 271–306 (1959).

    MathSciNet  MATH  Google Scholar 

  35. A. V. Zabrodin and G. P. Prokopov, “Method for the numerical modeling of plane unsteady flows of heat-conducting gas in three-temperature approximation,” Vopr. At. Nauki Tekh., Ser. Mat. Model. Fiz. Protsessov, No. 3, 3–16 (1998).

    Google Scholar 

  36. G. P. Prokopov, Preprint No. 66, IPM RAN (Keldysh Inst. of Applied Mathematics, Russian Academy of Sciences, Moscow, 2004).

    Google Scholar 

  37. N. Ya. Moiseev and E. A. Shestakov, “Solution of the Riemann problem in two-and three-temperature gas dynamics,” Comput. Math. Math. Phys. 55 (9), 1547–1553 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  38. W. Zhang, L. Howell, A. Almgren, A. Burrows, J. Dolence, and J. Bell, “Castro: A new compressible astrophysical solver. III: Multigroup radiation hydrodynamics,” The Astrophys. J. Suppl. Ser. 204 (7), 1–27 (2013).

    Google Scholar 

  39. A. N. Kraiko, Theoretical Gas Dynamics (Torus, Moscow, 2010) [in Russian].

    Google Scholar 

  40. A. A. Samarskii and Yu. P. Popov, Finite Difference Methods for Problems in Gas Dynamics (Nauka, Moscow, 1975) [in Russian].

    Google Scholar 

  41. N. Ya. Moiseev, “Consistent approximation in godunov-type difference schemes for one-dimensional problems of gas dynamics,” Comput. Math. Math. Phys. 36 (1), 125–126 (1996).

    Google Scholar 

  42. R. P. Fedorenko, “The application of difference schemes of high accuracy to the numerical solution of hyperbolic equations,” USSR Comput. Math. Math. Phys. 2 (6), 1355–1365 (1962).

    Article  MATH  Google Scholar 

  43. N. Ya. Moiseev and I. Yu. Silant’eva, “High-order accurate difference schemes for solving gasdynamic equations by the Godunov method with antidiffusion,” Comput. Math. Math. Phys. 49 (5), 827–841 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  44. N. Ya. Moiseev, “High-order accurate implicit running schemes,” Comput. Math. Math. Phys. 51 (5), 862–875 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  45. K. A. Bagrinovskii and S. K. Godunov, “Finite-difference schemes for multidimensional problems,” Dokl. Akad. Nauk SSSR 115, 431–433 (1957).

    MathSciNet  MATH  Google Scholar 

  46. A. S. Vershinskaya, D. S. Netsvetaev, A. V. Urakova, and A. A. Shestakov, “On a test RGD problem concerning the compression of a layered system with allowance for radiative transfer in various approzimations,” Abstracts of the Zababakhin Scientific Conference (Ross. Fed. Yadern. Tsentr Vseross. Nauchn. Issled. Inst. Tekh. Fiz., Snezhinsk, 2014).

    Google Scholar 

  47. A. S. Antonov, Parallel Programming with Use of OpenMP Technology (Mosk. Univ., Moscow, 2009) [in Russian].

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. All-Russia Research Institute of Technical Physics, Russian Federal Nuclear Center, Snezhinsk, Chelyabinsk oblast, 456770, Russia

    N. Ya. Moiseev

Authors
  1. N. Ya. Moiseev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. Ya. Moiseev.

Additional information

Original Russian Text © N.Ya. Moiseev, 2017, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2017, Vol. 57, No. 2, pp. 303–315.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Moiseev, N.Y. Modified method of splitting with respect to physical processes for solving radiation gas dynamics equations. Comput. Math. and Math. Phys. 57, 306–317 (2017). https://doi.org/10.1134/S0965542517020117

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0965542517020117

Keywords

Search

Navigation