Advertisement

Numerical Stability of the Discretized Initial Boundary Value Problems of the String–Diffusion Systems of Equations (SDSE)

  • Oleg Novik
  • Feodor Smirnov
  • Maxim Volgin
Chapter

Abstract

The string–diffusion systems of equations (SDSE), being formulated correctly from a physical viewpoint (see Chap.  9), are not partial differential equations (PDEs; scalar or vector ones) for which the type may be determined in the frame of the classical typing of PDEs of mathematical physics, because this typing was formed when fields of different natures were investigated mostly without taking account of their interaction. But precursors to earthquakes (one of the main subjects of this book) are generated because of interactions of physical fields of different natures. Therefore, the finite difference and Galerkin methods for initial boundary value problems of SDSE are considered in this chapter.

Keywords

Interaction of physical fields of different natures Stability of an algorithm 

References

  1. Adams, R., & Fournier, J. (2003). Sobolev spaces (p. 320). Amsterdam: Elsevier.Google Scholar
  2. Babich, V. M. (2009). Asymptotic methods in short-wavelength diffraction theory (alpha science series on wave phenomena). Oxford: Alpha Science.Google Scholar
  3. Beckenbach, E., & Bellman, R. (1975). Introduction to inequalities. New York: Random House.Google Scholar
  4. Egorov, Y. V. (1984). Linear differential equations of the main type. Moscow: Nauka (In Russian: Линейные дифференциальные уравнения главного типа. Москва, Наука. 359 с.).Google Scholar
  5. Knopoff, L. (1955). The interaction between elastic wave motion and a magnetic field in electrical conductors. Journal of Geophysical Research, 60(4), 441–446.CrossRefGoogle Scholar
  6. Ladyjenskaya, O. A., Solonnikov, V. A., & Uraltseva, N. N. (1967). Linear and quasilinear equations of parabolic type. Translations of mathematical monographs (Vol. 23). Providence: American Mathematical Society. (Translated from Russian, Moscow: Izdat. Nauka 1964).Google Scholar
  7. Ladyjenskaya, O. A. (1969). The mathematical theory of viscous incompressible flow. New York: Gordon and Breach, Science Publications. (Translated from Russian, Moscow: Gos. Izdat. Fiz-Mat Lit. 1961).Google Scholar
  8. Leony, G. (2009). A first course in Sobolev spaces (p. 607). Providence: American Mathematical Society.Google Scholar
  9. Maugin, G. (2013). Continuum mechanics trough the twentieth century. New York: Springer.CrossRefGoogle Scholar
  10. Mikhaylovskaya, I. B., Novik, O. B. (1979). The Cauchy problem in the class of growing functions for a non-hypoelliptic system of equations which is not hyperbolic. (In Russian: Михайловская И.Б., Новик О.Б. Задача Коши в классе растущих функций для негипоэллиптической системы уравнений, не являющейся гиперболической //Сибирский математический журнал. Деп. ВИНИТИ. 1979.Ч. 1. № 2104–79. 22 с. Ч. 2. № 2105–79. 22 с.).Google Scholar
  11. Mikhaylovskaya, I. B., & Novik, O. B. (1995). Net approximation of a conservative–dissipative system of equations. (In Russian: Михайловская И.Б., Новик О.Б. Сеточное приближение консервативно-диссипативной системы уравнений // «Фундаментальные и прикладные проблемы механики деформируемых сред и конструкций». Программа Госкомвуза РФ по высшему образованию: Науч. труды. Вып. 2. Изд-во Нижегородского гос. ун-та, 1995. Стр. 21–33).Google Scholar
  12. Novik, O. B. (1969). The Cauchy problem for the system of PDEs including the hyperbolic and parabolic operators. Journal of Computational Mathematics and Mathematical Physics, 1, 122–136 (In Russian).Google Scholar
  13. Novik, O. B. (1983a). Interaction of the geothermic field with the field of deformations of a layer. Geothermic investigations in Middle Asia and Kasahstan. Abstracts of reports of all-union conference, Ashgabat (In Russian).Google Scholar
  14. Novik, O. B. (1983b). Mathematical description of non-stationary fields of deformations and temperature. In Proceedings of high educational institutes (Известия вузов) (p. 24). Ser. Geology and Prospecting. Deposited in VINITI. № 6788-83 (In Russian).Google Scholar
  15. Novik, O. B. (1988). Finite difference schemes for the problems of thermo-mechanical effect in the oceanic lithosphere: collection of scientific papers. In Geothermic investigations on the bottoms of the water areas (pp. 89–97). Мoscow: Nauka (Наука) (In Russian).Google Scholar
  16. Novik, O. B. (1993). Galerkin method for the 2D system of equations of magneto-thermo-elasticity. (In Russian: Новик О.Б. Метод Галеркина для двумерной системы уравнений магнито-термо-упругости: Сб. «Фундаментальные и прикладные проблемы механики деформируемых сред и конструкций». Вып. 1. Н. Новгород, 1993. Стр. 117–124).Google Scholar
  17. Novik, O.B. (1994a). Galerkin method for the 3D non-linear system of equations of magneto-thermo-elasticity. (In Russian: Новик О.Б. Метод Галеркина для трехмерной нелинейной системы уравнений магнито-термо-упругости. Доклады РАН. 1994. Т. 334. № 2. Стр. 100–102. The Journal is translated and published in English).Google Scholar
  18. Novik, O. B. (1994b). Seismogenic magneto-thermo-elastic effect. (In Russian: Новик О.Б. Сейсмогенный магнитотермоупругий эффект. Доклады PAH. 1994б. Т. 338. № 2. Стр. 238–241. The Journal is translated and published in English).Google Scholar
  19. Novik, O. B. (1995). Interaction of heat-transfer in the active lithosphere zones with other geophysical fields. (In Russian: Новик О.Б. Взаимодействие теплопереноса в активных зонах литосферы с другими геофизическими полями. Международный симпозиум «Проблемы геотермальной энергии»: Сб. основных докладов. 1995. Стр. 56–73).Google Scholar
  20. Rihtmayer, R. D., & Morton, K. W. (1994). Difference methods for initial-value problems (2nd ed.). Malabar: Krieger.Google Scholar
  21. Sobolev, S. L., & Vaskevich, V. (1997). Theory of cubature formulas. New York: Springer.CrossRefGoogle Scholar
  22. Yang, E. H. (1988). On some new discrete generalizations of Gronwall’s inequality. Journal of Mathematical Analysis and Applications, 129(2), 505–516.CrossRefGoogle Scholar
  23. Wang, L. (2003). On Korn’s inequality. Journal of Computational Mathematics, 21(3), 321–324.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Oleg Novik
    • 1
  • Feodor Smirnov
    • 1
  • Maxim Volgin
    • 1
  1. 1.IZMIRAN of the Russian Academy of SciencesMoscowRussia

Personalised recommendations