Acceleration of transition to stationary mode for solutions to a system of viscous gas dynamics. (English. Russian original) Zbl 1428.35416

Mosc. Univ. Math. Bull. 74, No. 2, 55-61 (2019); translation from Vestn. Mosk. Univ., Ser. I 74, No. 2, 14-21 (2019).
Summary: Explicit formulas for the initial data stabilization algorithm for the stationary solution are obtained by zero approximation method for the semi-implicit difference scheme approximating a system of equations for the dynamics of a one-dimensional viscous barotropic gas. The spectrum of the corresponding linearized system on the stationary solution is studied and theoretical convergence estimates are obtained. Numerical experiments for the nonlinear problem are carried out to confirm the efficiency of the method and to reflect the dependence of the stabilization rate on the parameters of the original problem and the algorithm.


35Q35 PDEs in connection with fluid mechanics
76N15 Gas dynamics (general theory)
35B35 Stability in context of PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35P05 General topics in linear spectral theory for PDEs
Full Text: DOI


[1] Zhukov, K. A.; Kornev, A. A.; Popov, A. V., Acceleration of the Process of Entering Stationary Mode for Solutions of a Linearized System of Viscous Gas Dynamics. I, II, Vestn. Mosk. Univ., Matem. Mekhan., 1, 26, (2018) · Zbl 1397.35236
[2] A. G. Kulikovskii, N. V. Pogorelov, and A. Yu. Semenov, Mathematical Issues of Numerical Solution of Hyperbolic Systems of Equations (Nauka, Moscow, 2001) [in Russian]. · Zbl 0965.35001
[3] Lebedev, V. I., Difference Analogies of Orthogonal Decompositions, Basic Differential Operators, and Some Boundary Value Problems of Mathematical Physics I, II, Zh. Vychisl. Matem. Matem. Fiz., 3, 449, (1964)
[4] Imranov, F. B.; Kobelkov, G. M.; Sokolov, A. G., Finite Difference Scheme for Barotropic Gas Equations, Doklady Rus. Akad. Nauk, 478, 388, (2018) · Zbl 1444.76074
[5] Zvyagin, A. V.; Kobelkov, G. M.; Lozhnikov, M. A., On Some Finite Difference Scheme for Gas Dynamics Equations, 15, (2018) · Zbl 1402.76089
[6] Chizhonkov, E. V., Numerical Aspects of One Stabilization Method, Russ. J. Numer. Anal. Math. Modelling, 18, 363, (2003) · Zbl 1058.65095
[7] A. A. Samarskii and E. S. Nikolaev, Solution Methods for Grid Equations (Nauka, Moscow, 1978) [in Russian].
[8] N. S. Bakhvalov, N. P. Zhidkov, and G. M. Kobel’kov, Numerical Methods (BINOM, Moscow, 2012) [in Russian]. · Zbl 0638.65001
[9] Milyutin, S. V.; Chizhonkov, E. V., Two Methods of Approximate Projection onto Stable Manifold, Vychisl. Metody Programm., 8, 177, (2007)
[10] Chizhonkov, E. V., Projection Operators for Numerical Stabilization, Vychisl. Metody Programm., 5, 161, (2004)
[11] Fursikov, A. V., Stabilizability of a Quasi-Linear Parabolic Equation by Means of a Boundary Control with Feedback, Matem. Sborn., 192, 115, (2001) · Zbl 1019.93047
[12] Kornev, A. A., Classification of Methods of Approximate Projecting onto Stable Manifold, Doklady Rus. Akad. Nauk, 400, 736, (2005)
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