Advertisement

Doklady Mathematics

, Volume 98, Issue 2, pp 458–463 | Cite as

On Conditions for L2-Dissipativity of Linearized Explicit QGD Finite-Difference Schemes for One-Dimensional Gas Dynamics Equations

  • A. A. Zlotnik
  • T. A. Lomonosov
Mathematics

Abstract

An explicit two-level in time and spatially symmetric finite-difference scheme approximating the 1D quasi-gasdynamic system of equations is studied. The scheme is linearized about a constant solution, and new necessary and sufficient conditions for the L2-dissipativity of solutions to the Cauchy problem are derived, including, for the first time, the case of a nonzero background velocity and depending on the Mach number. It is shown that the condition on the Courant number can be made independent of the Mach number. The results provide a substantial development of the well-known stability analysis of the linearized Lax–Wendroff scheme.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    B. N. Chetverushkin, Kinetic Schemes and Quasi-Gasdynamic System of Equations (MAKS, Moscow, 2004; CIMNE, Barcelona, 2008).Google Scholar
  2. 2.
    T. G. Elizarova, Quasi-Gas Dynamic Equations (Nauchnyi Mir, Moscow, 2007; Springer-Verlag, Berlin, 2009).CrossRefzbMATHGoogle Scholar
  3. 3.
    Yu. V. Sheretov, Continuum Dynamics under Spatiotemporal Averaging (Regulyarnaya i Khaoticheskaya Dinamika, Moscow, 2009) [in Russian].Google Scholar
  4. 4.
    A. A. Zlotnik and B. N. Chetverushkin, Comput. Math. Math. Phys. 48 (3), 420–446 (2008).MathSciNetCrossRefGoogle Scholar
  5. 5.
    A. A. Zlotnik, Dokl. Math. 81 (2), 312–316 (2010).MathSciNetCrossRefGoogle Scholar
  6. 6.
    A. A. Sukhomozgii and Yu. V. Sheretov, in Applications of Functional Analysis in Approximation Theory (Tver. Gos. Univ., Tver, 2013), pp. 48–60 [in Russian].Google Scholar
  7. 7.
    A. Zlotnik and T. Lomonosov, in Differential and Difference Equations with Applications (Springer, Cham, 2018), pp. 635–647. https://arxiv.org/abs/1803.09899.CrossRefGoogle Scholar
  8. 8.
    S. K. Godunov and V. S. Ryaben’kii, Difference Schemes: An Introduction to the Underlying Theory (Nauka, Moscow, 1977; North-Holland, Amsterdam, 1987).Google Scholar
  9. 9.
    R. D. Richtmyer and K. W. Morton, Difference Methods for Initial-Value Problems (Wiley, New York, 1967).zbMATHGoogle Scholar
  10. 10.
    A. A. Zlotnik, Comput. Math. Math. Phys. 52 (7), 1060–1071 (2012).MathSciNetCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.National Research University Higher School of EconomicsMoscowRussia

Personalised recommendations