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Solutions of Multidimensional Hydrodynamic Evolution Equations Using the Fast Legendre Transformation

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Mathematical Modeling and Supercomputer Technologies (MMST 2022)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 1750))

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Abstract

The numerical solution of multidimensional nonlinear evolution equations, in which the field depends on several spatial coordinates, is considered. The work was based on the Fast Legendre Transform algorithm. A model for the numerical calculation of the processes of surface growth and the evolution of a two-dimensional velocity field, implemented in the MATLAB software environment, is presented. With the help of this model, the growth of regular surfaces and surfaces with a random initial shape (two-dimensional Gaussian noise with zero mean and unit variance) was considered, and the level lines and lines of discontinuities of the growing surfaces were also shown. The result of numerical simulation of a localized random velocity field with an initial field in the form of two-dimensional Gaussian noise with zero mean and unit variance is presented.

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References

  1. Burgers, J.M.: Mathematical examples illustrating relations occurring in the theory of turbulent fluid motion. In: Nieuwstadt, F.T.M., Steketee, J.A. (eds.). Selected Papers of J.M. Burgers, pp. 281–334. Springer, Dordrecht (1995). https://doi.org/10.1007/978-94-011-0195-0_10

  2. Gurbatov, S.N., Rudenko, O.V., Saichev, A.I.: Waves and Structures in Nonlinear Nondispersive Media. 1st edn. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-23617-4

  3. Saichev, A.I., Woyczyński, W.A.: Distributions in the Physical and Engineering Sciences, vol. 2, 1st edn. Birkhäuser, New York (2013). https://doi.org/10.1007/978-0-8176-4652-3

  4. Rudenko, O.V., Soluyan, S.I.: Theoretical Foundations of Nonlinear Acoustics. Consultants Bureau, New York (1977)

    Book  MATH  Google Scholar 

  5. Saichev, A.I., Woyczyński, W.A.: Distributions in the Physical and Engineering Sciences, vol. 3, 1st edn. Birkhäuser, Cham (2008). https://doi.org/10.1007/978-3-319-97958-8

  6. Mehran, K.M., Giorgio, P.G., Zhang, Y.-C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56(9), 889 (1986). https://doi.org/10.1103/PhysRevLett.56.889

    Article  MATH  Google Scholar 

  7. Gurbatov, S.N., Saichev, A.I., Shandarin, S.F.: The large-scale structure of the Universe in the frame of the model equation of non-linear diffusion. Mon. Not. R. Astron. Soc. 236(2), 385–402 (1989)

    Article  MATH  Google Scholar 

  8. Vergassola, M., Dubrulle, B., Frisch, U., Noullez, A.: Burgers’ equation, Devil’s staircases and the mass distribution for large-scale structures. Astron. Astrophys. 289, 325–356 (1994)

    Google Scholar 

  9. Gurbatov, S.N., Saichev, A.I., Shandarin, S.F.: Large-scale structure of the Universe. The Zeldovich approximation and the adhesion model. Phys. Usp. 55(3), 223–249 (2012). https://doi.org/10.3367/UFNe.0182.201203a.0233

    Article  Google Scholar 

  10. Noullez, A., Vergassola, M.: A fast Legendre transform algorithm and applications to the adhesion model. J. Sci. Comput. 9, 259–281 (1994). https://doi.org/10.1007/BF01575032

    Article  MathSciNet  MATH  Google Scholar 

  11. Spivak, A.E., Gurbatov, S.N., Demin, I.Yu.: Numerical solution of the Burgers equation in the case of vanishingly small viscosity using the fast Legendre transformation. Memoirs of the Faculty of Physics (2022). (in press)

    Google Scholar 

  12. Murugan, S.D., Frisch, U., Nazarenko, S., Besse, N., Ray, S.S.: Suppressing thermalization and constructing weak solutions in truncated inviscid equations of hydrodynamics: lessons from the Burgers equation. Phys. Rev. Res. 2(3), 033202 (2020). https://doi.org/10.1103/PhysRevResearch.2.033202

    Article  Google Scholar 

  13. Bequin, P.: Burgers equation and convexification method: propagation of wave packets and narrowband noise. J. Acoust. Soc. Am. 151(2), 1223 (2022). https://doi.org/10.1121/10.0009578

    Article  Google Scholar 

  14. Gurbatov, S.N., Moshkov, A., Noullez, A.: The evolution of anisotropic structures and turbulence in the multi-dimensional Burgers equation. Phys. Rev. E 81(4), 046312 (2010). https://doi.org/10.1103/PhysRevE.81.046312

    Article  MathSciNet  Google Scholar 

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Acknowledgement

This work has been supported by the grants the Russian Science Foundation, RSF 19-12-00256.

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Authors and Affiliations

  1. Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod, 603950, Russia

    A. E. Spivak, S. N. Gurbatov & I. Yu. Demin

Authors
  1. A. E. Spivak
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  2. S. N. Gurbatov
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  3. I. Yu. Demin
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Correspondence to A. E. Spivak .

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Editors and Affiliations

  1. Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, Russia

    Dmitry Balandin

  2. Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, Russia

    Konstantin Barkalov

  3. Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, Russia

    Iosif Meyerov

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Spivak, A.E., Gurbatov, S.N., Demin, I.Y. (2022). Solutions of Multidimensional Hydrodynamic Evolution Equations Using the Fast Legendre Transformation. In: Balandin, D., Barkalov, K., Meyerov, I. (eds) Mathematical Modeling and Supercomputer Technologies. MMST 2022. Communications in Computer and Information Science, vol 1750. Springer, Cham. https://doi.org/10.1007/978-3-031-24145-1_8

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  • DOI: https://doi.org/10.1007/978-3-031-24145-1_8

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-24144-4

  • Online ISBN: 978-3-031-24145-1

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