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Solution to a Linearized System of Two-Dimensional Dynamics of Viscous Gas

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Abstract

A linear system of partial differential equations approximately describing the dynamics of small perturbations of a nonstationary viscous barotropic gas in a neighborhood of the steady state is considered in the paper. Analytic formulas for the solution are obtained for initial conditions of special type, and the asymptotics of the rate of convergence to the stationary solution is studied. Similar assertions are proved for a finite-difference approximation of the original problem constructed on grids of Lebedev. In addition, the presence of analytical formulas for the solution allows explaining why a perturbation of velocity jump type decreased significantly better than that for a pressure jump. The results form a basis for studying the problem of asymptotic stabilization of solutions to two-dimensional equations of gas dynamics with dissipation terms.

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REFERENCES

  1. V. I. Lebedev, ‘‘Difference analogues of orthogonal decompositions, basic differential operators and some boundary problems of mathematical physics. I,’’ USSR Comput. Math. Math. Phys. 4 (3), 69–92 (1964). https://doi.org/10.1016/0041-5553(64)90240-X

    Article  MATH  Google Scholar 

  2. V. I. Lebedev, ‘‘Difference analogues of orthogonal decompositions, basic differential operators and some boundary problems of mathematical physics. II,’’ USSR Comput. Math. Math. Phys. 4 (4), 36–50 (1964). https://doi.org/10.1016/0041-5553(64)90003-5

    Article  Google Scholar 

  3. K. A. Zhukov, A. A. Kornev, and A. V. Popov, ‘‘Acceleration of the process of entering stationary mode for solutions of a linearized system of viscous gas dynamics. I,’’ Moscow Univ. Math. Bull. 73, 24–29 (2018). https://doi.org/10.3103/S0027132218010047

    Article  MathSciNet  MATH  Google Scholar 

  4. K. A. Zhukov, A. A. Kornev, and A. V. Popov, ‘‘Acceleration of the process of entering stationary mode for solutions of a linearized system of viscous gas dynamics. II,’’ Moscow Univ. Math. Bull. 73, 85–89 (2018). https://doi.org/10.3103/S0027132218030014

    Article  MathSciNet  MATH  Google Scholar 

  5. K. A. Zhukov, A. A. Kornev, M. A. Lozhnikov, and A. V. Popov, ‘‘Acceleration of transition to stationary mode for solutions to a system of viscous gas dynamics,’’ Moscow Univ. Math. Bull. 74, 55–61 (2019). https://doi.org/10.3103/S0027132219020037

    Article  MathSciNet  MATH  Google Scholar 

  6. E. V. Chizhonkov, ‘‘Numerical aspects of one stabilization method,’’ Russ. J. Numer. Anal. Math. Modell. 18, 363–376 (2003). https://doi.org/10.1515/rnam.2003.18.5.363

    Article  MathSciNet  MATH  Google Scholar 

  7. A. V. Fursikov, ‘‘Stabilizability of a quasi-linear parabolic equation by means of a boundary control with feedback,’’ Sb. Math. 192, 593–639 (2001). https://doi.org/10.1070/SM2001v192n04ABEH000560

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Chair of Computational Mathematics, Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia

    A. A. Kornev & V. S. Nazarov

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  1. A. A. Kornev
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  2. V. S. Nazarov
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Correspondence to A. A. Kornev or V. S. Nazarov.

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Translated by E. Oborin

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Kornev, A.A., Nazarov, V.S. Solution to a Linearized System of Two-Dimensional Dynamics of Viscous Gas. Moscow Univ. Math. Bull. 77, 209–216 (2022). https://doi.org/10.3103/S0027132222050023

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