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Zh. Vychisl. Mat. Mat. Fiz., 2008, Volume 48, Number 3, Pages 445–472 (Mi zvmmf169)  

This article is cited in 51 scientific papers (total in 52 papers)

Parabolicity of the quasi-gasdynamic system of equations, its hyperbolic second-order modification, and the stability of small perturbations for them

A. A. Zlotnika, B. N. Chetverushkinb

a Department of Applied Mathematics, Russian State Social University, ul. V. Pika 4, Moscow, 129226, Russia
b Institute of Mathematical Modeling, Russian Academy of Sciences, Miusskaya pl. 4a, Moscow, 125047, Russia

Abstract: Criteria (necessary and sufficient conditions) for the Petrovskii parabolicity of the quasi-gasdynamic system of equations with an improved description of heat conduction are derived. A modified quasi-gasdynamic system containing second derivatives with respect to both spatial and time variables is proposed. Necessary and sufficient conditions for its hyperbolicity are deduced. For both systems, the stability of small perturbations against a constant background is analyzed and estimates that are uniform on an infinite time interval are given for relative perturbations in the Cauchy problem and the initial-boundary value problem for the corresponding linearized systems. Similar results are also established in the barotropic case with the general equation of state $p=p(\rho)$.

Key words: gas dynamics, quasi-gasdynamic system of equations, second-order parabolic and hyperbolic systems, principal symbol, stability of small perturbations, symmetrization, Fourier transform.

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English version:
Computational Mathematics and Mathematical Physics, 2008, 48:3, 420–446

Bibliographic databases:

UDC: 519.633
Received: 23.07.2007

Citation: A. A. Zlotnik, B. N. Chetverushkin, “Parabolicity of the quasi-gasdynamic system of equations, its hyperbolic second-order modification, and the stability of small perturbations for them”, Zh. Vychisl. Mat. Mat. Fiz., 48:3 (2008), 445–472; Comput. Math. Math. Phys., 48:3 (2008), 420–446

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Zlotnik A.A., Chetverushkin B.N., “Parabolicity of the quasi-gasdynamic system of equations, hyperbolicity of one of its modifications, and stability of small perturbations for them”, Dokl. Math., 77:1 (2008), 143–148  mathnet  mathscinet  zmath  isi  elib  elib
    2. A. A. Zlotnik, “Some properties of the equations governing a two-dimensional quasi-gasdynamic model of traffic flows”, Comput. Math. Math. Phys., 49:2 (2009), 363–372  mathnet  crossref  mathscinet  zmath  isi
    3. Zlotnik A.A., Gavrilin V.A., “O kriteriyakh parabolichnosti kvazigidrodinamicheskoi sistemy uravnenii v sluchae realnogo gaza”, Vestn. Mosk. energeticheskogo in-ta, 2009, no. 6, 116–126
    4. Chetverushkin B.N., “High-performance computing: fundamental problems in industrial applications”, Parallel, distributed and grid computing for engineering, Computational Science Engineering and Technology Series, 21, 2009, 369–388  crossref  isi
    5. A. A. Zlotnik, “On stability of small perturbations for a modified two-dimensional quasi-gasdynamic model of traffic flows”, Math. Models Comput. Simul., 2:6 (2010), 776–781  mathnet  crossref  mathscinet  zmath
    6. Zlotnik A.A., “Quasi-gasdynamic system of equations with general equations of state”, Dokl. Math., 81:2 (2010), 312–316  crossref  mathscinet  zmath  isi  elib  elib  scopus
    7. A. A. Zlotnik, “O kvazigazodinamicheskoi sisteme uravnenii s obschimi uravneniyami sostoyaniya i istochnikom tepla”, Matem. modelirovanie, 22:7 (2010), 53–64  mathnet  mathscinet
    8. A. A. Zlotnik, “Energy equalities and estimates for barotropic quasi-gas-dynamic and quasi-hydrodynamic systems of equations”, Comput. Math. Math. Phys., 50:2 (2010), 310–321  mathnet  crossref  mathscinet  adsnasa  isi
    9. A. A. Davydov, B. N. Chetverushkin, E. V. Shilnikov, “Simulating flows of incompressible and weakly compressible fluids on multicore hybrid computer systems”, Comput. Math. Math. Phys., 50:12 (2010), 2157–2165  mathnet  crossref  adsnasa
    10. Zlotnik A.A., “Linearized stability of equilibrium solutions to the quasi-gasdynamic system of equations”, Dokl. Math., 82:2 (2010), 811–815  crossref  mathscinet  zmath  isi  elib  elib  scopus
    11. O. V. Bulatov, T. G. Elizarova, “Regularized shallow water equations and an efficient method for numerical simulation of shallow water flows”, Comput. Math. Math. Phys., 51:1 (2011), 160–173  mathnet  crossref  mathscinet  isi
    12. Zlotnik A., Gavrilin V., “On quasi-gasdynamic system of equations with general equations of state and its application”, Math. Model. Anal., 16:4 (2011), 509–526  crossref  mathscinet  zmath  isi  elib  scopus
    13. Elizarova T.G., Bulatov O.V., “Regularized shallow water equations and a new method of numerical simulation of the open channel flows”, Comput. & Fluids, 46:1 (2011), 206–211  crossref  mathscinet  zmath  isi  scopus
    14. Elizarova T.G., Zlotnik A.A., Nikitina O.V., “Modelirovanie odnomernykh techenii melkoi vody na osnove regulyarizovannykh uravnenii”, Preprinty IPM im. M.V. Keldysha, 2011, no. 33, 1–36  elib
    15. T. G. Elizarova, A. A. Zlotnik, O. V. Nikitina, “Modelirovanie odnomernykh techenii melkoi vody na osnove regulyarizovannykh uravnenii”, Preprinty IPM im. M. V. Keldysha, 2011, 033, 36 pp.  mathnet
    16. Zlotnik A.A., “Spatial discretization of one-dimensional quasi-gasdynamic systems of equations and the entropy and energy balance equations”, Dokl. Math., 86:1 (2012), 464–468  crossref  mathscinet  zmath  isi  elib  elib  scopus
    17. Boris N. Chetverushkin, “Resolution limits of continuous media models and their mathematical formulations”, Math. Models Comput. Simul., 5:3 (2013), 266–279  mathnet  crossref  mathscinet
    18. A. A. Zlotnik, “O postroenii kvazigazodinamicheskikh sistem uravnenii i barotropnoi sistemy s potentsialnoi massovoi siloi”, Matem. modelirovanie, 24:4 (2012), 65–79  mathnet  mathscinet
    19. A. A. Zlotnik, “Prostranstvennaya diskretizatsiya odnomernoi barotropnoi kvazigazodinamicheskoi sistemy uravnenii i uravnenie energeticheskogo balansa”, Matem. modelirovanie, 24:10 (2012), 51–64  mathnet  mathscinet
    20. Chetverushkin B.N., Gulin A.V., “Explicit schemes and numerical simulation using ultrahigh-performance computer systems”, Dokl. Math., 86:2 (2012), 681–683  crossref  mathscinet  zmath  isi  elib  elib  scopus
    21. A. A. Ilin, Yu. G. Rykov, “Ob odnom modelnom uravnenii s malym parametrom pri starshei proizvodnoi po vremeni, voznikayuschem pri analize nekotorykh kvazigazodinamicheskikh sistem uravnenii”, Preprinty IPM im. M. V. Keldysha, 2012, 012, 9 pp.  mathnet
    22. A. A. Ilin, Yu. G. Rykov, “Ob odnoi modelnoi sisteme s malym parametrom pri starshei proizvodnoi po vremeni, voznikayuschem pri analize nekotorykh kvazigazodinamicheskikh sistem uravnenii”, Preprinty IPM im. M. V. Keldysha, 2012, 075, 9 pp.  mathnet
    23. Ducomet B. Zlotnik A., “On a Regularization of the Magnetic Gas Dynamics System of Equations”, Kinet. Relat. Mod., 6:3 (2013), 533–543  crossref  mathscinet  zmath  isi  elib  scopus
    24. A. I. Aptekarev, A. L. Afendikov, V. M. Buchstaber, S. K. Godunov, A. B. Zhizhchenko, V. V. Kozlov, A. I. Maslov, V. P. Platonov, V. F. Tishkin, L. D. Faddeev, “Boris Nikolaevich Chetverushkin (on his 70th birthday)”, Russian Math. Surveys, 69:2 (2014), 387–392  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    25. T. G. Elizarova, O. V. Bulatov, “Chislennyi algoritm resheniya regulyarizovannykh uravnenii melkoi vody na nestrukturirovannykh setkakh”, Preprinty IPM im. M. V. Keldysha, 2014, 021, 27 pp.  mathnet
    26. V. A. Gavrilin, A. A. Zlotnik, “On spatial discretization of the one-dimensional quasi-gasdynamic system of equations with general equations of state and entropy balance”, Comput. Math. Math. Phys., 55:2 (2015), 264–281  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    27. B. N. Chetverushkin, V. I. Savelev, “Kineticheskie modeli i vysokoproizvoditelnye vychisleniya”, Preprinty IPM im. M. V. Keldysha, 2015, 079, 31 pp.  mathnet
    28. A. A. Zlotnik, “On conservative spatial discretizations of the barotropic quasi-gasdynamic system of equations with a potential body force”, Comput. Math. Math. Phys., 56:2 (2016), 303–319  mathnet  crossref  crossref  isi  elib
    29. O. V. Bulatov, T. G. Elizarova, “Regularized shallow water equations for numerical simulation of flows with a moving shoreline”, Comput. Math. Math. Phys., 56:4 (2016), 661–679  mathnet  crossref  crossref  mathscinet  isi  elib
    30. V. A. Balashov, A. A. Zlotnik, E. B. Savenkov, “Issledovanie barotropnoi kvazigidrodinamicheskoi modeli dvukhfaznoi smesi s uchetom poverkhnostnykh effektov”, Preprinty IPM im. M. V. Keldysha, 2016, 089, 25 pp.  mathnet  crossref
    31. A. A. Ilin, Yu. G. Rykov, “O blizosti traektorii dlya modelnykh kvazigazodinamicheskikh uravnenii. Lineinyi sluchai”, Preprinty IPM im. M. V. Keldysha, 2016, 090, 14 pp.  mathnet  crossref
    32. M. A. Istomina, T. G. Elizarova, “Kvazigazodinamicheskii algoritm dlya polyarnoi sistemy koordinat i primer chislennogo modelirovaniya neustoichivostei v akkretsionnom diske”, Preprinty IPM im. M. V. Keldysha, 2016, 092, 25 pp.  mathnet  crossref
    33. Zlotnik A. Gavrilin V., “On a Conservative Finite-Difference Method for 1D Shallow Water Flows Based on Regularized Equations”, Mathematical Problems in Meteorological Modelling, Mathematics in Industry, 24, ed. Batkai A. Csomos P. Farago I. Horanyi A. Szepszo G., Springer Int Publishing Ag, 2016, 3–18  crossref  mathscinet  zmath  isi
    34. Ilyin A.A. Rykov Yu.G., “On the closeness of trajectories for model quasi-gasdynamic equations”, Dokl. Math., 94:2 (2016), 543–546  crossref  mathscinet  zmath  isi  elib  scopus
    35. T. G. Elizarova, A. A. Zlotnik, M. A. Istomina, “O dvumernom chislennom KGD modelirovanii spiralno-vikhrevykh struktur v akkretsionnykh gazovykh diskakh”, Preprinty IPM im. M. V. Keldysha, 2017, 001, 30 pp.  mathnet  crossref
    36. A. A. Zlotnik, “Entropy-conservative spatial discretization of the multidimensional quasi-gasdynamic system of equations”, Comput. Math. Math. Phys., 57:4 (2017), 706–725  mathnet  crossref  crossref  mathscinet  isi  elib
    37. T. E. Moiseev, E. E. Myshetskaya, V. F. Tishkin, “O blizosti reshenii nevozmuschënnykh i giperbolizirovannykh uravnenii teploprovodnosti dlya razryvnykh nachalnykh dannykh”, Preprinty IPM im. M. V. Keldysha, 2017, 086, 15 pp.  mathnet  crossref
    38. Chetverushkin B.N. Zlotnik A.A., “On Some Properties of Multidimensional Hyperbolic Quasi-Gasdynamic Systems of Equations”, Russ. J. Math. Phys., 24:3 (2017), 299–309  crossref  mathscinet  zmath  isi  scopus
    39. Zlotnik A.A. Chetverushkin B.N., “Entropy Balance For the One-Dimensional Hyperbolic Quasi-Gasdynamic System of Equations”, Dokl. Math., 95:3 (2017), 276–281  crossref  mathscinet  zmath  isi  scopus
    40. Chetverushkin B.N. Zlotnik A.A., “On a Hyperbolic Perturbation of a Parabolic Initial-Boundary Value Problem”, Appl. Math. Lett., 83 (2018), 116–122  crossref  mathscinet  zmath  isi  scopus
    41. Saburin D.S. Elizarova T.G., “Modelling the Azov Sea Circulation and Extreme Surges in 2013-2014 Using the Regularized Shallow Water Equations”, Russ. J. Numer. Anal. Math. Model, 33:3 (2018), 173–185  crossref  mathscinet  zmath  isi  scopus
    42. Zlotnik A., “On the Energy Dissipative Spatial Discretization of the Barotropic Quasi-Gasdynamic and Compressible Navier–Stokes Equations in Polar Coordinates”, Russ. J. Numer. Anal. Math. Model, 33:3 (2018), 199–210  crossref  mathscinet  zmath  isi  scopus
    43. Elizarova T.G. Zlotnik A.A. Istomina M.A., “Hydrodynamical Aspects of the Formation of Spiral-Vortical Structures in Rotating Gaseous Disks”, Astron. Rep., 62:1 (2018), 9–18  crossref  isi  scopus
    44. B. N. Chetverushkin, M. V. Yakobovskii, “Vychislitelnye algoritmy i arkhitektura sistem vysokoi proizvoditelnosti”, Preprinty IPM im. M. V. Keldysha, 2018, 052, 12 pp.  mathnet  crossref  elib
    45. B. N. Chetverushkin, “Giperbolicheskaya kvazigazodinamicheskaya sistema”, Matem. modelirovanie, 30:2 (2018), 81–98  mathnet  elib
    46. B. N. Chetverushkin, N. D'Ascenzo, A. V. Saveliev, V. I. Saveliev, “Kinetic model and magnetogasdynamics equations”, Comput. Math. Math. Phys., 58:5 (2018), 691–699  mathnet  crossref  crossref  isi  elib
    47. B. N. Chetverushkin, A. V. Saveliev, V. I. Saveliev, “A quasi-gasdynamic model for the description of magnetogasdynamic phenomena”, Comput. Math. Math. Phys., 58:8 (2018), 1384–1394  mathnet  crossref  crossref  isi  elib
    48. Zlotnik A.A. Lomonosov T.A., “On Conditions For l-2-Dissipativity of Linearized Explicit Qgd Finite-Difference Schemes For One-Dimensional Gas Dynamics Equations”, Dokl. Math., 98:2 (2018), 458–463  crossref  mathscinet  zmath  isi  scopus
    49. Zlotnik A. Lomonosov T., “On Conditions For Weak Conservativeness of Regularized Explicit Finite-Difference Schemes For 1D Barotropic Gas Dynamics Equations”, Differential and Difference Equations With Applications, Springer Proceedings in Mathematics & Statistics, 230, ed. Pinelas S. Caraballo T. Kloeden P. Graef J., Springer, 2018, 635–647  crossref  mathscinet  zmath  isi  scopus
    50. Zlotnik A., “On l-2-Dissipativity of Linearized Explicit Finite-Difference Schemes With a Regularization on a Non-Uniform Spatial Mesh For the 1D Gas Dynamics Equations”, Appl. Math. Lett., 92 (2019), 115–120  crossref  mathscinet  zmath  isi  scopus
    51. Zlotnik A. Lomonosov T., “Verification of An Entropy Dissipative Qgd-Scheme For the 1D Gas Dynamics Equations”, Math. Model. Anal., 24:2 (2019), 179–194  crossref  isi
    52. B. N. Chetverushkin, A. E. Lutskii, V. P. Osipov, “Zakony sokhraneniya i kompaktnaya kvazigazodinamicheskaya sistema”, Matem. modelirovanie, 31:12 (2019), 21–32  mathnet  crossref  elib
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