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DESIGNING AND DETAILING OF BUILDING SYSTEMS. MECHANICS IN CIVIL ENGINEERING

Numerical investigation of the Carleman system

Vestnik MGSU 6/2015
  • Vasil’eva Ol’ga Aleksandrovna - Moscow State University of Civil Engineering (National Research University) (MGSU) Candidate of Physical and Mathematical Sciences, Associate Professor, Department of Higher Mathematics, Moscow State University of Civil Engineering (National Research University) (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .

Pages 7-15

In the article the Cauchy problem of the Carleman equation is considered. The Carleman system of equations is a model problem of the kinetic theory of gases. It is a discrete kinetic model of one-dimensional gas consisting of identical monatomic molecules. The molecules can have one of two speeds, which have equal values and opposite directions. This system of the equations is quasi-linear hyperbolic system of partial differential equations. There is no analytic solution for this problem in general case. So, the numerical investigation of the Cauchy problem of the Carleman system solution is very important.The paper presents and discusses the results of the numerical investigation of the Cauchy problem for the studied system solution with periodic initial conditions. The dependence of the stabilization time of the solution and the time dependence of energy exchange from small parameter are obtained.The second point of the paper is numerical investigation of the solution of the Cauchy problem with non-periodic initial conditions. The solution stabilization to the equilibrium state is obtained. The solution stabilization time is compared with stabilization time in periodic case.The final point of the paper is numerical investigation of the Cauchy problem with stationary normal processes as initial conditions. The solution to this problem is two stationary stochastic processes for any fixed value of time variable. As a rule, the practical interest is not a stochastic solution but its statistical characteristics. The stochastic solution realization is presented and discussed. The dependence of the mathematical expectation of the solution deviation modulus from equilibrium state is obtained. It demonstrates the process of the solution stabilization.

DOI: 10.22227/1997-0935.2015.6.7-15

References
  1. Bol'tsman L. Izbrannye trudy [Selected Works]. Moscow, Nauka Publ., 1984, 590 p. (Classics of Science) (In Russian)
  2. Godunov S.K., Sultangazin U.M. O diskretnykh modelyakh kineticheskogo uravneniya Bol’tsmana [On Discrete Models of Boltzmann Kinetic Equation]. Uspekhi Matematicheskihk Nauk [The Success of Mathematical Sciences]. 1971, vol. 26, no. 3 (159), pp. 3—51.(In Russian)
  3. Radkevich E.V. O diskretnykh kineticheskikh uravneniyakh [On Discrete Kinetic Equations]. Doklady Akademii nauk [Reports of the Academy of Sciences]. 2012, vol. 447, no. 4, pp. 369—373. (In Russian)
  4. Euler N., Steeb W.-H. Painleve Test and Discrete Boltzmann Equations. Aust. J. Phys. 1989, vol. 42, pp. 1—10. DOI: http://dx.doi.org/10.1071/PH890001.
  5. Radkevich E.V. The Existence of Global Solutions to the Cauchy Problem for Discrete Kinetic Equations. Journal of Mathematical Science. 2012, vol. 181, no. 2, pp. 232— 280. DOI: http://dx.doi.org/10.1007/s10958-012-0683-9.
  6. Radkevich E.V., Vasil’eva O.A., Dukhnovskii S.A. Local Equilibrium of the Carleman Equation. Journal of Mathematical Science. 2015, vol. 207, no. 2, pp. 296—323.
  7. Radkevich E.V. O povedenii na bol'shikh vremenakh resheniy zadachi Koshi dlya dvumernogo diskretnogo kineticheskogo uravneniya [The Behavior of Cauchy Problem Solutions for Two-Dimensional Discrete Kinetic Equation at Large Times]. Sovremennaya matematika. Fundamental'nye napravleniya [Contemporary Mathematics. Fundamental Directions]. 2013, vol. 47, pp. 108—139. (In Russian)
  8. Adzhiev S.Z., Amosov S.A., Vedenyapin V.V. Odnomernye diskretnye modeli kineticheskikh uravneniy dlya smesey [One Dimensional Discrete Models of Kinetic Equations for Mixtures]. Zhurnal vychislitel’noy matematiki i matematicheskoy fiziki [Journal of Computational Mathematics and Mathematical Physics]. 2004, vol. 44, no. 3, pp. 553—558. (In Russian)
  9. Il’in O.V. Izuchenie sushchestvovaniya resheniy i ustoychivosti kineticheskoy sistemy Karlemana [Investigating the Existence of Solutions and Stability of Carleman Kinetic System]. Zhurnal vychislitel’noy matematiki i matematicheskoy fiziki [Journal of Computational Mathematics and Mathematical Physics]. 2007, vol. 47, no. 12, pp. 2076—2087. (In Russian)
  10. Aristov V., Ilyin O. Kinetic Model of the Spatio-Temporal Turbulence. Phys. Let. A. 2010, vol. 374, no. 43, pp. 4381—4384. DOI: http://dx.doi.org/10.1016/j.physleta.2010.08.069.
  11. Illner R., Reed M.C., Neunzert H. The Decay of Solutions of the Carleman Model. Math. Methods Appl. Sci. 1981, vol. 3 (1), pp. 121—127. DOI: http://dx.doi.org/10.1002/mma.1670030110.
  12. Illner R., Reed M.C. Decay the Equilibrium for the Carleman Model in a Box. SIAM J. Appl. Math. 1984, vol. 44, no. 6, pp. 1067—1075. DOI: http://dx.doi.org/10.1137/0144076.
  13. Aristov V.V. Direct Methods for Solving the Boltzmann Equation and Study of Nonequilibrium Flows. Kluwer Academic Publishing, 2001, 312 p.
  14. Radkevich E.V. Matematicheskie voprosy neravnovesnykh protsessov [Mathematical Problems of Nonequilibrium Processes]. Novosibirsk, T. Rozhkovskaya Publ., 2007, 300 p. (White Series in Mathematics and Physics. Vol. 4). (In Russian)
  15. Il’in O.V. Statsionarnye resheniya kineticheskoy modeli Broduella [Stationary Solutions of the Kinetic Broadwell Model]. Teoreticheskaya i matematicheskaya fizika [Theoretical and Mathematical Physics]. 2012, vol. 170, no. 3, pp. 481—488. (In Russian)
  16. Frishter L.Yu. Analiz napryazhenno-deformirovannogo sostoyaniya v vershine pryamougol’nogo klina [Analysis of Stress-strain State on Top of a Rectangular Wedge]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2014, no. 5, pp. 57—62. (In Russian)
  17. Radkevich E.V. The Existence of Global Solutions to the Cauchy Problem for Discrete Kinetic Equations (Non-Periodic Case). Journal of Mathematical Science. 2012, vol. 184, no. 4, pp. 524—556. DOI: http://dx.doi.org/10.1007/s10958-012-0879-z.
  18. Vasil’eva O.A. Issledovanie nekotorykh veroyatnostnykh kharakteristik resheniya zadachi Koshi dlya uravneniya Byurgersa-Khaksli [Investigation of Some Statistical Characteristics of the Cauchy Problem for Burgers-Huxley Equation Solution]. Trudy MAI [Works of Moscow Aviation Institute]. 2014, no. 78. Available at: http://www.mai.ru/science/trudy/published.php?ID=53684. (In Russian)
  19. Vasil’eva O.A. Programmnyy modul’ CORFUN 1.2.-2 [The Program Unit CORFUN 1.2.-2]. Matematika. Komp’yuter. Obrazovanie : trudy XVIII Mezhdunarodnoy konferentsii [Mathematics. Computer. Education : Works of the 18th International Conference]. 2011, no. 18, p. 193. (In Russian)
  20. Vasil’eva O.A. Issledovanie veroyatnostnykh kharakteristik resheniya uravneniya Byurgersa-Khaksli [Investigation of Statistical Characteristics of the Burgers-Huxley Equation Solution]. Matematika. Komp’yuter. Obrazovanie : trudy XXII Mezhdunarodnoy konferentsii [Mathematics. Computer. Education : Works of the 22nd International Conference]. 2015, no. 22, p. 130. (In Russian)

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NUMERICAL SOLUTION OF THE GODUNOV - SULTANGAZIN SYSTEM OF EQUATIONS. PERIODIC CASE

Vestnik MGSU 4/2016
  • Vasil’eva Ol’ga Aleksandrovna - Moscow State University of Civil Engineering (National Research University) (MGSU) Candidate of Physical and Mathematical Sciences, Associate Professor, Department of Higher Mathematics, Moscow State University of Civil Engineering (National Research University) (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .

Pages 27-35

The Cauchy problem of the Godunov - Sultangazin system of equations with periodic initial conditions is considered in the article. The Godunov - Sultangazin system of equations is a model problem of the kinetic theory of gases. It is a discrete kinetic model of one-dimensional gas consisting of identical monatomic molecules. The molecules can have one of three speeds. So, there are three groups of molecules. The molecules of the first two groups have the speeds equal in values and opposite in directions. The molecules of the third group have zero speed. The considered mathematical model has a number of properties of Boltzmann equation. This system of the equations is a quasi-linear system of partial differential equations. There is no analytic solution for this problem in the general case. So, numerical investigation of the Cauchy problem of the Godunov - Sultangazin system is very important. The finite-difference method of the first order is used for numerical investigation of the Cauchy problem of the Godunov - Sultangazin system of equations. The paper presents and discusses the results of numerical investigation of the Cauchy problem for the studied system solution with periodic initial condition. The dependence of the time of stabilization of the Cauchy problem solution of Godunov - Sultangazin system of equations from the decreasing parameter of system are obtained. The paper presents the dependence of time of energy exchange from the decreasing parameter. The solution stabilization to the equilibrium state is obtained. The stabilization time of Godunov - Sultangazin system solution is compared to the stabilization time of Carleman system solution in periodic case. The results of numerical investigation are in good agreement with the theoretical results obtained previously.

DOI: 10.22227/1997-0935.2016.4.27-35

References
  1. Boltzmann L. Izbrannye trudy [Selected Works]. Moscow, Nauka Publ., 1984, 590 p. (In Russian)
  2. Godunov S.K., Sultangazin U.M. O diskretnykh modelyakh kineticheskogo uravneniya Bol’tsmana [On Discreet Models of Kinetic Boltzmann Equation]. UMN [Success of Mathematical Sciences]. 1974, vol. XXVI, no. 3 (159), pp. 3—51. (In Russian)
  3. Radkevich E.V. O diskretnykh kineticheskikh uravneniyakh [On Discreet Kinetic Equations]. Doklady Akademii nauk [Reports of the Academy of Sciences]. 2012, vol. 447, no. 4, p. 369. (In Russian)
  4. Radkevich E.V. The Existence of Global Solutions to the Cauchy Problem for Discrete Kinetic Equations. Journal of Mathematical Science. 2012, vol. 181, no. 2, pp. 232—280. DOI: http://dx.doi.org/10.1007/s10958-012-0683-9.
  5. Radkevich E.V. The Existence of Global Solutions to the Cauchy Problem for Discrete Kinetic Equations II. Journal of Mathematical Science. 2012, vol. 181, no. 5, pp. 701—750. DOI: http://dx.doi.org/10.1007/s10958-012-0711-9.
  6. Radkevich E.V., Vasil’eva O.A., Dukhnovskii S.A. Local Equilibrium of the Carleman Equation. Journal of Mathematical Science. 2015, vol. 207, no. 2, pp. 296—323. DOI: http://dx.doi.org/10.1007/s10958-015-2373-x.
  7. Radkevich E.V. O povedenii na bol’shikh vremenakh resheniy zadachi Koshi dlya dvumernogo kineticheskogo uravneniya [On the Behavior of Cauchy Problem Solutions at Large Times for Two-Dimensional Kinetic Equation]. Sovremennaya matematika. Fundamental’nye napravleniya [Modern Mathematics. Fundamental Directions]. 2013, vol. 47, pp. 108—139. (In Russian)
  8. Radkevich E.V. The Existence of Global Solutions to the Cauchy Problem for Discrete Kinetic Equations (Non-Periodic Case). Journal of Mathematical Science. 2012, vol. 184, no. 4, pp. 524—556. DOI: http://dx.doi.org/10.1007/s10958-012-0879-z.
  9. Adzhiev S.Z., Amosov S.A., Vedenyapin V.V. Odnomernye diskretnye modeli kineticheskikh uravneniy dlya smesey [One-Dimensional Discrete Models of Kinetic Equations for Mixes]. Zhurnal vychislitel’noy matematiki i matematicheskoy fiziki [Computational Mathematics and Mathematical Physics]. 2004, vol. 44, no. 3, pp. 553—555. (In Russian)
  10. Aristov V.V. Direct Methods for Solving the Boltzmann Equation and Study of Nonequilibrium Flows. Fluid Mechanics and its Applications. Kluwer Academic Publishing, 2001, vol. 60, 312 p. DOI: http://dx.doi.org/10.1007/978-94-010-0866-2.
  11. Vasil’eva O. Some Results of Numerical Investigation of the Carleman System. Procedia Engineering 24th. “XXIV R-S-P Seminar — Theoretical Foundation of Civil Engineering, TFoCE 2015”. 2015, vol. 111, pp. 834—838. DOI: http://dx.doi.org/10.1016/j.proeng.2015.07.154.
  12. Vasil’eva O.A. Chislennoe issledovanie sistemy uravneniy Karlemana [Numerical Investigation of the Carleman System]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2015, no. 6, pp. 7—15. (In Russian)
  13. Radkevich E.V. Matematicheskie voprosy neravnovesnykh protsessov [Mathematical Problems of Nonequilibrium Processes]. Novosibirsk, T. Rozhkovskaya Publ., 2007, 300 p. (In Russian)
  14. Medvedeva N.A. Vazhnost’ individual’nogo podkhoda pri obuchenii vysshey matematike v tekhnicheskom vuze [The Importance of Individual Approach in Teaching Higher Mathematics at Technical Universities]. Stroitel’stvo: nauka i obrazovanie [Construction: Science and Education]. 2015, no. 4. Paper 1. Available at: http://nso-journal.ru. (In Russian)
  15. Bobyleva T.N. Opredelenie rezonansnykh chastot osesimmetrichnykh kolebaniy pologo shara s ispol’zovaniem uravneniy dvizheniya trekhmernoy teorii uprugosti [Determination of Resonant Frequencies of Axisymmetric Oscillations of a Hollow Ball Using of the Equations of Motion of Three-Dimensional Elasticity Theory]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2015, no. 7, pp. 25—32. (In Russian)
  16. Bobyleva T.N. Opredelenie rezonansnykh chastot osesimmetrichnykh kolebaniy uprugogo izotropnogo pologo shara na osnove uravneniy dvizheniya Lame [Determination of Resonant Frequencies of Axisymmetric Vibrations of Elastic Isotropic Hollow Ball on the Basis of Lame Motion Equation]. Estestvennye i tekhnicheskie nauki [Natural and Technical Sciences]. 2015, no. 3 (81), pp. 46—49. (In Russian)
  17. Bobyleva T.N. Rasprostranenie osesimmetrichnykh elektrouprugikh voln v krugovykh p’ezokeramicheskikh tsilindrakh s osevoy polyarizatsiey [Propagation of Axisymmetric Electroelastic Waves in a Circular Piezoceramic Cylinders with Axial Polarization]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2010, no. 4—3, pp. 16—20. (In Russian)
  18. Frishter L.Yu. Otsenki resheniya odnorodnoy ploskoy zadachi teorii uprugo-sti v okrestnosti neregulyarnoy tochki granitsy [Evaluations of the Solution to the Homogeneous Two-Dimensional Problem of the Theory of Elasticity in the Vicinity of an Irregular Point of the Border]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2012, no. 2, pp. 20—24. (In Russian)
  19. Frishter L.Yu. Analiz napryazhenno-deformirovannogo sostoyaniya v vershine pryamougol’nogo klina [Analysis of Stress-strain State on Top of a Rectangular Wedge]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2014, no. 5, pp. 57—62. (In Russian)
  20. Vasil’eva O.A. Chislennoe issledovanie diskretnykh kineticheskikh uravneniy [Numerical Investigation of Discrete Kinetic Equations]. Matematika. Komp’yuter. Obrazovanie : trudy XXIII Mezhdunarodnoy konferentsii [Mathematics. Computer. Education : Works of the 23rd International Conference]. 2016, no. 23, 192 p. (In Russian)

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