Numerical Simulation and its Results

  • Vladimir DanilovEmail author
  • Roman Gaydukov
  • Vadim Kretov
Part of the Heat and Mass Transfer book series (HMT)


In the first part of this chapter, a numerical algorithm for solving the phase field system is presented with application to the real field emission nanocathode. The second part of this chapter contains the results of numerical simulations. In the third part of this chapter, we present an algorithm for introducing a liquid phase nucleus in the presented mathematical model of heat transfer in nanocathodes.


  1. 1.
    Chapman, B., Jost, G., van der Pas, R.: Using OpenMP: Portable Shared Memory Parallel Programming. The MIT Press (2008)Google Scholar
  2. 2.
    Cook, S.: Cuda Programming: A Developer’s Guide to Parallel Computing with GPUs. Elsevier (2013)Google Scholar
  3. 3.
    Danilov, V.G., Gaydukov, R.K., Kretov, V.I., Rudnev, V.Y.: Modelling of liquid nuclei generation for field-emission silicon nanocathode. IEEE Trans. Electron Devices 61(12), 4232–4239 (2014)ADSCrossRefGoogle Scholar
  4. 4.
    Danilov, V.G., Omel’yanov, G.A., Radkevich, E.V.: Hugoniot-type conditions and weak solutions to the phase-field system. Eur. J. Appl. Math. 10, 55–77 (1999)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Dyzhev, N.A., Gudkova, S.A., Makhiboroda, M.A., Fedirko V, A.: Investigation of emussion properties of silicon cathodes of different geometry. In: Bykov, D.V. (Ed.) Vacuum Science and Technics, Material of XII Scientific-Technical Conference with Participation of Foreign Specialists, pp. 221–224. MIEM, Moscow (2005). (in Russian)Google Scholar
  6. 6.
    Glazov, V.M., Chizhevskaia, S.N., Glagoleva, N.N.: Liquid Semiconductors. Springer (1969)Google Scholar
  7. 7.
    Iyengar, S., Jain, R.: Numerical Methods. New Age International Ltd (2009)Google Scholar
  8. 8.
    Kolmogorov, A.N.: On the statistical theory of metal crystallization. Izvestiya Akademii Nauk SSSR. Ser. Matematicheskaya 1(3), 355–359 (1937). (in Russian)Google Scholar
  9. 9.
    Omel’yanov, G.A., Rudnev, V.Y.: Interaction of free boundaries in the modified Stefan problem. Nonlinear Phenom. Complex Syst. 7(3), 227–237 (2004)MathSciNetGoogle Scholar
  10. 10.
    Rauber, T., Rünger, G.: Parallel Programming For Multicore and Cluster System. Springer (2010)Google Scholar
  11. 11.
    Samarskii, A.A.: The Theory of Difference Schemes. Marcel Dekker Inc. (2001)Google Scholar
  12. 12.
    Ward, M.J., Reyna, L.G.: Resolving weak internal layer interactions for the ginzburg-landau equation. Eur. J. Appl. Math. 5, 495–523 (1994)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  • Vladimir Danilov
    • 1
    Email author
  • Roman Gaydukov
    • 1
  • Vadim Kretov
    • 1
  1. 1.National Research University Higher School of EconomicsMoscowRussia

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