Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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, II,” Vestn. Mosk. Univ., Matem. Mekhan., No. 1, 26 (2018); No. 3, 3 (2018).
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].
V. I. Lebedev, “Difference Analogies of Orthogonal Decompositions, Basic Differential Operators, and Some Boundary Value Problems of Mathematical Physics I, II,” Zh. Vychisl. Matem. Matem. Fiz. 3 (4), 449 (1964); 4 (4), 649 (1964).
F. B. Imranov, G. M. Kobelkov, and A. G. Sokolov, “Finite Difference Scheme for Barotropic Gas Equations,” Doklady Rus. Akad. Nauk 478 (4), 388 (2018) [Doklady Math. 97 (1), 58 (2018)].
A. V. Zvyagin, G. M. Kobelkov, and M. A. Lozhnikov, “On Some Finite Difference Scheme for Gas Dynamics Equations,” Vestn. Mosk. Univ., Matem. Mekhan., No. 4, 15 (2018).
E. V. Chizhonkov, “Numerical Aspects of One Stabilization Method,” Russ. J. Numer. Anal. Math. Modelling 18 (5), 363 (2003).
A. A. Samarskii and E. S. Nikolaev, Solution Methods for Grid Equations (Nauka, Moscow, 1978) [in Russian].
N. S. Bakhvalov, N. P. Zhidkov, and G. M. Kobel’kov, Numerical Methods (BINOM, Moscow, 2012) [in Russian].
S. V. Milyutin and E. V. Chizhonkov, “Two Methods of Approximate Projection onto Stable Manifold,” Vychisl. Metody Programm. 8, 177 (2007).
E. V. Chizhonkov, “Projection Operators for Numerical Stabilization,” Vychisl. Metody Programm. 5, 161 (2004).
A. V. Fursikov, “Stabilizability of a Quasi-Linear Parabolic Equation by Means of a Boundary Control with Feedback,” Matem. Sborn. 192 (4), 115 (2001) [Sbornik: Math. 192 (4), 593 (2001)].
A. A. Kornev, “Classification of Methods of Approximate Projecting onto Stable Manifold,” Doklady Rus. Akad. Nauk 400 (6), 736 (2005).
Author information
Authors and Affiliations
Corresponding authors
About this article
Cite this article
Zhukov, K.A., Kornev, A.A., Lozhnikov, M.A. et al. 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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S0027132219020037