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International Conference on Differential & Difference Equations and Applications

ICDDEA 2019: Differential and Difference Equations with Applications pp 53–67Cite as

On Enlarged Sufficient Conditions for \(L^2\)-Dissipativity of Linearized Explicit Schemes with Regularization for 1D Gas Dynamics Systems of Equations

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 333))

Abstract

We study an explicit two-level in time and three-point symmetric in space finite-difference scheme for 1D barotropic and full gas dynamics systems of equations. The scheme is a linearization at a constant background solution (with an arbitrary velocity) of finite-difference schemes with general viscous regularization. We enlarge recently proved sufficient conditions (on the Courant-like number) for \(L^2\)-dissipativity in the Cauchy problem for the schemes by deriving new bounds for the commutator of matrices of viscous and convective terms. We deal with the case of a kinetic regularization in more detail and specify sufficient conditions in this case where the mentioned matrices are closely connected. Importantly, these new sufficient conditions rapidly tend to the known necessary ones as the Mach number grows. Also several forms of setting a regularization parameter are considered.

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Acknowledgements

The study was supported by the Russian Foundation for Basic Research, project nos. 19-11-00169 and 18-01-00587.

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Authors and Affiliations

  1. Department of Mathematics at Faculty of Economic Sciences, National Research University Higher School of Economics, Pokrovskii bd. 11, 109028, Moscow, Russia

    Alexander Zlotnik

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  1. Alexander Zlotnik
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Correspondence to Alexander Zlotnik .

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Editors and Affiliations

  1. Department of Exact Sciences and Engineering, Military Academy, Lisbon, Portugal

    Sandra Pinelas

  2. Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN, USA

    John R. Graef

  3. Mathematik und Didaktik, Katholische Universität Eichstätt, Eichstätt, Bayern, Germany

    Stefan Hilger

  4. Huazhong University of Science and Technology, Wuhan, Hubei, China

    Peter Kloeden

  5. Department of Electrical and Computer Engineering, Democritus University of Thrace, Xanthi, Xanthi, Greece

    Christos Schinas

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Zlotnik, A. (2020). On Enlarged Sufficient Conditions for \(L^2\)-Dissipativity of Linearized Explicit Schemes with Regularization for 1D Gas Dynamics Systems of Equations. In: Pinelas, S., Graef, J.R., Hilger, S., Kloeden, P., Schinas, C. (eds) Differential and Difference Equations with Applications. ICDDEA 2019. Springer Proceedings in Mathematics & Statistics, vol 333. Springer, Cham. https://doi.org/10.1007/978-3-030-56323-3_5

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