Mathematical Models and Computer Simulations

, Volume 1, Issue 1, pp 113–123 | Cite as

Sufficient stability conditions in the calculations of steady supersonic flows using the marching technique and time-dependent flows with account for viscosity

  • V. G. Grudnitskii
Article
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Abstract

The sufficient conditions for the stability and monotonicity in calculating supersonic steady flows by means of the marching technique are derived. The sufficient stability conditions are also obtained for constructing the solutions of time-dependent conservation laws with account for viscosity by explicit difference schemes. With increase in the viscosity coefficient, the conditions derived go over continuously from the hyperbolic to the parabolic constraints on the time step.

Keywords

Euler Equation Discontinuous Solution Godunov Scheme Gasdynamic Parameter Explicit Difference Scheme 
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References

  1. 1.
    V. G. Grudnitskii, “Generalized Characteristics for the System of Euler Equations and Their Application to the Construction of Difference Schemes,” Mat. Model., 4(12), 45–48 (1992).MathSciNetGoogle Scholar
  2. 2.
    V. G. Grudnitskii, “Generalized Characteristics for the System of Euler Equations,” in Algorithms for the Numerical Investigation of Discontinuous Solutions (Russian Academy of Sciences, Computer Center, Moscow, 1993) [in Russian], pp. 191–203.Google Scholar
  3. 3.
    V. G. Grudnitskii, “Generalized Characteristics and Sufficient Stability Condition for the Euler Equations with Discontinuous Solutions,” Mat. Model., 9(12), 121–125 (1997).MathSciNetGoogle Scholar
  4. 4.
    V. G. Grudnitskii, “Sufficient Stability Condition in Explicitly Constructing the Discontinuous Solutions of the System of Euler Equations,” Dokl. Ross. Akad. Nauk, 362, 298–299 (1998).MathSciNetGoogle Scholar
  5. 5.
    V. G. Grudnitskii, “Sufficient Stability Condition in the Multidimensional Calculations of the Time-Dependent Discontinuous Solutions of the Euler Equations,” Mat. Model., 12(1), 65–77 (2000).CrossRefMathSciNetGoogle Scholar
  6. 6.
    V. G. Grudnitskii and P. V. Plotnikov, “Generalized Characteristics and Sufficient Stability Condition in Constructing the Discontinuous Solutions of the System of Euler Equations,” in News in Numerical Modeling (2000) [in Russian], pp. 148–164.Google Scholar
  7. 7.
    V. G. Grudnitskiy, “Sufficient Conditions of Stability for Discontinuous Solutions of the Euler Equations,” Comp. Fluid Dyn. J., 10, 334–337, 2001.Google Scholar
  8. 8.
    V. G. Grudnitskii, “Direct Generalized Characteristic Method for Calculating Discontinuous Solutions of the Conservation Laws of Gas Dynamics,” Mat. Model., 16(1), 90–96 (2004).MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    V. G. Grudnitskii, “Sufficient Stability Conditions for the Godunov Scheme,” Mat. Model., 17(12), 119–128 (2005).CrossRefMathSciNetGoogle Scholar
  10. 10.
    V. G. Grudnitskii, “Sufficient Stability Conditions in Calculating the Discontinuous Solutions of the Time-Dependent Conservation Laws in Curvilinear Coordinates, in the Presence of Right Sides,” Mat. Model., 18(10), 76–80 (2006).Google Scholar
  11. 11.
    P. D. Lax, “Weak Solutions of Nonlinear Hyperbolic Equations and Their Numerical Computation,” Comm. Pure Appl. Math., 7, 159–193 (1954).MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    S. K. Godunov, “A Difference Method for Calculating Shock Waves,” Usp. Mat. Nauk, 12(1), 176–177 (1957).MATHMathSciNetGoogle Scholar
  13. 13.
    V. S. Ryaben’kii and A. F. Filippov, Stability of Difference Equations (Gostekhizdat, Moscow, 1956) [in Russian].Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2009

Authors and Affiliations

  • V. G. Grudnitskii
    • 1
  1. 1.Dorodnitsyn Computer CenterRussian Academy of SciencesMoscowRussia

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