Advertisement

Journal of Scientific Computing

, Volume 61, Issue 3, pp 513–532 | Cite as

A Semi-Lagrangian Method for 3-D Fokker Planck Equations for Stochastic Dynamical Systems on the Sphere

  • A. RothEmail author
  • A. Klar
  • B. Simeon
  • E. Zharovsky
Article

Abstract

In this paper, we consider stochastic dynamical systems on the sphere and the associated Fokker–Planck equations. A semi-Lagrangian method combined with a Finite Volume discretization of the sphere is presented to solve the Fokker–Planck equation. The method is applied to a typical problem in fiber dynamics and textile production. The numerical results are compared to explicit solutions and Monte-Carlo solutions.

Keywords

Fokker–Planck equation SDE Fiber lay-down Semi-Lagrange method Finite volume method 

Notes

Acknowledgments

This work has been supported by Deutsche Forschungsgemeinschaft (DFG), KL 1105/18-1 and by Bundesministerium für Bildung und Forschung (BMBF), Verbundprojekt OPAL.

References

  1. 1.
    Adcroft, A., Campin, J., Hill, C., Marshall, J.: Implementation of an atmosphere–ocean general circulation model on the expanded spherical cube. Mon. Wea. Rev. 132, 2845–2863 (2004)CrossRefGoogle Scholar
  2. 2.
    Bonilla, L., Götz, T., Klar, A., Marheineke, N., Wegener, R.: Hydrodynamic limit of a Fokker–Planck equation describing fiber lay-down processes. SIAM J. Appl. Math. 68(3), 648–665 (2007)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Degond, P., Motsch, S.: Continuum limit of self-driven particles with orientation interaction. Math. Models Methods Appl. Sci. 18, 1193–1215 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Degond, P., Appert-Rolland, C., Moussaid, M., Pettre, J., Theraulaz, G.: A hierarchy of heuristic-based models of crowd dynamics. http://arxiv.org/abs/1304.1927
  5. 5.
    Dolbeault, J., Klar, A., Mouhot, C., Schmeiser, C.: Hypocoercivity and a Fokker–Planck equation for fiber lay-down. Appl. Math. Res. Exp. 2013(2), 165–175 (2013)Google Scholar
  6. 6.
    Douglas, J., Huang, C., Pereira, F.: The modified method of characteristics with adjusted advection. Numer. Math. 83, 353–369 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Douglas, J., Russell, T.F.: Numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite elements or finite differences. SIAM J. Numer. Anal. 19, 871–885 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Götz, T., Klar, A., Marheineke, N., Wegener, R.: A stochastic model and associated Fokker–Planck equation for the fiber lay-down process in nonwoven production processes. SIAM J. Appl. Math. 67(6), 1704–1717 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Grothaus, M., Klar, A.: Ergodicity and rate of convergence for a non-sectorial fiber lay-down process. SIAM J. Math. Anal. 40(3), 968–983 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Grothaus, M., Klar, A., Maringer, J., Stilgenbauer, P.: Geometry, mixing, properties and hypocoercivity of a degenerate diffusion arising in technical textile industry. http://arxiv.org/abs/1203.4502
  11. 11.
    Kageyama, A., Sato, T.: Yin-Yang grid : an overset grid in spherical geometry. Geochem. Geophys. Geosyst. 5, Q09005 (2004)CrossRefGoogle Scholar
  12. 12.
    Klar, A., Reuterswärd, P., Seaïd, M.: A semi-Lagrangian method for a Fokker–Planck equation describing fiber dynamics. J. Sci. Comp. 38(3), 349–367 (2009)CrossRefzbMATHGoogle Scholar
  13. 13.
    Klar, A., Marheineke, N., Wegener, R.: Hierarchy of mathematical models for production processes of technical textiles. ZAMM 89(12), 941–961 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Klar, A., Maringer, J., Wegener, R.: A 3D model for fiber lay-down processes in non-woven production processes. Math. Models Methods Appl. Sci. 22, 9 (2012)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kolb, M., Savov, M., Wuebker, A.: (Non)ergodicity of a degenerate diffusion modeling the fiber lay down process. SIAM J. Math. Anal. 45(1), 113 (2012)Google Scholar
  16. 16.
    Leveque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge (2002)CrossRefzbMATHGoogle Scholar
  17. 17.
    Oeksendahl, B.: Stochastic Differential equations. Springer, Berlin (2005)Google Scholar
  18. 18.
    Sadourny, R., Arakawa, A., Mintz, Y.: Integration of the nondivergent barotropic vorticity equation with an icosahedral–hexagonal grid for the sphere. Mon. Wea. Rev. 96, 351–356 (1968)CrossRefGoogle Scholar
  19. 19.
    Seaïd, M.: On the quasi-monotone modified method of characteristics for transport-diffusion problems with reactive sources. Comp. Methods Appl. Math. 2, 186–210 (2002)zbMATHGoogle Scholar
  20. 20.
    Stroock, D.W.: On the growth of stochastic integrals. Z.Wahr. verw.Geb. 18, 340–344 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Vicsek, T., Czirok, A., Ben-Jacob, E., Cohen, I., Shochet, O.: Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 75, 1226–1229 (1995)CrossRefGoogle Scholar
  22. 22.
    Wachspress, E.: A Rational Finite Element Basis. Academic Press, New York (1975)zbMATHGoogle Scholar
  23. 23.
    Zharovsky, E., Simeon, B.: A space-time adaptive approach to orientation dynamics in particle laden flow. Procedia Comput. Sci. 1, 791–799 (2010)CrossRefGoogle Scholar
  24. 24.
    Zharovski, E., Moosaie, A., LeDuc, A., Manhart, M., Simeon, B.: On the numerical solution of a convection–diffusion equation for particle orientation dynamics on geodesic grids. Appl. Numer. Math. 62(10), 15541566 (2012)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of MathematicsTechnische Universität KaiserslauternKaiserslauternGermany
  2. 2.Fraunhofer ITWMKaiserslauternGermany

Personalised recommendations