About this Journal Submit a Manuscript Table of Contents
International Journal of Differential Equations
Volume 2013 (2013), Article ID 314290, 11 pages
http://dx.doi.org/10.1155/2013/314290
Research Article

Embedded Zassenhaus Expansion to Splitting Schemes: Theory and Multiphysics Applications

Institute of Physics, Felix-Hausdorff-Street 6, 17489 Greifswald, Germany

Received 25 March 2013; Revised 5 August 2013; Accepted 19 August 2013

Academic Editor: Shuyu Sun

Copyright © 2013 Jürgen Geiser. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. R. E. Ewing, “Up-scaling of biological processes and multiphase ow in porous media,” in IIMA Volumes in Mathematics and Its Applications, vol. 295, pp. 195–215, Springer, New York, NY, USA, 2002.
  2. J. Geiser, Discretization and Simulation of Systems for Convection-Diffusion-Dispersion Reactions with Applications in Groundwater Contamination, Groundwater Modelling, Management and Contamination, Nova Science; Monograph, New York, NY, USA, 2008.
  3. N. Antonic, C. J. van Duijn, W. Jäger, and A. Mikelic, Multiscale Problems in Science and Technology: Challenges to Mathematical Analysis and Perspectives, Springer, Berlin, Germany, 2002.
  4. E. Weinan, Principle of Multiscale Modelling, Cambridge University Press, Cambridge, UK, 2011.
  5. S. Blanes and F. Casas, “On the necessity of negative coefficients for operator splitting schemes of order higher than two,” Applied Numerical Mathematics, vol. 54, no. 1, pp. 23–37, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  6. E. Hansen and A. Ostermann, “High order splitting methods for analytic semigroups exist,” BIT. Numerical Mathematics, vol. 49, no. 3, pp. 527–542, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  7. G. Strang, “On the construction and comparison of difference schemes,” SIAM Journal on Numerical Analysis, vol. 5, pp. 506–517, 1968. View at Publisher · View at Google Scholar · View at MathSciNet
  8. G. I. Marchuk, “Some applicatons of splitting-up methods to the solution of problems in mathematical physics,” Aplikace Matematiky, vol. 1, pp. 103–132, 1968.
  9. J. Geiser, “Computing exponential for iterative splitting methods: algorithms and applications,” Journal of Applied Mathematics, vol. 2011, Article ID 193781, 27 pages, 2011. View at Publisher · View at Google Scholar
  10. R. I. McLachlan and G. R. W. Quispel, “Splitting methods,” Acta Numerica, vol. 11, pp. 341–434, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  11. J. Geiser, G. Tanoğlu, and N. Gücüyenen, “Higher order operator splitting methods via Zassenhaus product formula: theory and applications,” Computers & Mathematics with Applications, vol. 62, no. 4, pp. 1994–2015, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  12. G. H. Weiss and A. A. Maradudin, “The Baker-Hausdorff formula and a problem in crystal physics,” Journal of Mathematical Physics, vol. 3, pp. 771–777, 1962. View at Publisher · View at Google Scholar · View at MathSciNet
  13. R. M. Wilcox, “Exponential operators and parameter differentiation in quantum physics,” Journal of Mathematical Physics, vol. 8, pp. 962–982, 1967. View at Publisher · View at Google Scholar · View at MathSciNet
  14. J. Geiser, “An iterative splitting approach for linear integro-differential equations,” Applied Mathematics Letters, vol. 26, no. 11, pp. 1048–1052, 2013.
  15. F. Casas and A. Murua, “An efficient algorithm for computing the Baker-Campbell-Hausdorff series and some of its applications,” Journal of Mathematical Physics, vol. 50, no. 3, Article ID 033513, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  16. J. Geiser, Iterative Splitting Methods for Differential Equations, Chapman & Hall/CRC Numerical Analysis and Scientific Computing, CRC Press, Boca Raton, Fla, USA, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  17. S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems, Teubner, Stuttgart, Germany, 1993.
  18. J. Geiser and G. Tanoglu, “Operator-splitting methods via the Zassenhaus product formula,” Applied Mathematics and Computation, vol. 217, no. 9, pp. 4557–4575, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  19. F. Bayen, “On the convergence of the Zassenhaus formula,” Letters in Mathematical Physics, vol. 3, no. 3, pp. 161–167, 1979. View at Publisher · View at Google Scholar · View at MathSciNet
  20. F. Casas, A. Murua, and M. Nadinic, “Efficient computation of the Zassenhaus formula,” Computer Physics Communications, vol. 183, no. 11, pp. 2386–2391, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  21. E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical Integration, vol. 31 of Springer Series in Computational Mathematics, Springer, Berlin, Germany, 2002. View at MathSciNet
  22. H. Yoshida, “Construction of higher order symplectic integrators,” Physics Letters A, vol. 150, no. 5–7, pp. 262–268, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  23. E. Faou, A. Ostermann, and K. Schratz, “Analysis of exponential splitting methods for inhomogeneous parabolic equations,” http://arxiv.org/abs/1212.5827.
  24. J. Geiser, “Discretization methods with embedded analytical solutions for convection-diffusion dispersion-reaction equations and applications,” Journal of Engineering Mathematics, vol. 57, no. 1, pp. 79–98, 2007. View at Publisher · View at Google Scholar · View at MathSciNet