• Balac, Stephane (UEB, Universite Europeenne de Bretagne, Universite de Rennes I)
  • Received : 2013.06.17
  • Accepted : 2013.10.15
  • Published : 2013.12.25


The Interaction Picture (IP) method is a valuable alternative to Split-step methods for solving certain types of partial differential equations such as the nonlinear Schr$\ddot{o}$dinger equation or the Gross-Pitaevskii equation. Although very similar to the Symmetric Split-step (SS) method in its inner computational structure, the IP method results from a change of unknown and therefore do not involve approximation such as the one resulting from the use of a splitting formula. In its standard form the IP method such as the SS method is used in conjunction with the classical 4th order Runge-Kutta (RK) scheme. However it appears to be relevant to look for RK scheme of higher order so as to improve the accuracy of the IP method. In this paper we investigate 5th order Embedded Runge-Kutta schemes suited to be used in conjunction with the IP method and designed to deliver a local error estimation for adaptive step size control.


  1. B.M. Caradoc-Davies. Vortex dynamics in Bose-Einstein condensate. PhD thesis, University of Otago (NZ), 2000.
  2. M.J. Davis. Dynamics in Bose-Einstein condensate. PhD thesis, University of Oxford (UK), 2001.
  3. S. W¨uster, T.E. Argue, and C.M. Savage. Numerical study of the stability of skyrmions in Bose-Einstein condensates. Phys. Rev. A, 72(4), 2005.
  4. R. Scott, C. Gardiner, and D. Hutchinson. Nonequilibrium dynamics: Studies of the reflection of Bose-Einstein condensates. Laser Phys., 17:527-532, 2007.
  5. C.N. Liu, G.G. Krishna, M. Umetsu, and S. Watanabe. Numerical investigation of contrast degradation of Bose-Einstein condensate interferometers. Phys. Rev. A, 79(1), 2009.
  6. J. Hult. A fourth-order Runge-Kutta in the Interaction Picture method for simulating supercontinuum generation in optical fibers. J. Lightwave Technol., 25(12):3770-3775, 2007.
  7. A. Heidt. Efficient adaptive step size method for the simulation of supercontinuum generation in optical fibers. J. Lightwave Technol., 27(18):3984-3991, 2009.
  8. A. Fernandez, S. Balac, A. Mugnier, F. Mahe, R. Texier-Picard, T. Chartier, and D. Pureur. Numerical simulation of incoherent optical wave propagation in nonlinear fibers. To appear in Eur. Phys. J. - Appl. Phys., 2013.
  9. J.C. Butcher. Numerical methods for ordinary differential equations. John Wiley and Sons, 2008.
  10. E. Hairer, S.P. Norsett, and G. Wanner. Solving ordinary differential equations I: nonstiff problems. Springer-Verlag, 1993.
  11. M. Crouzeix and A. Mignot. Analyse numerique des equations differentielles. Masson, Paris, 1984.
  12. J.S. Townsend. A modern approach to quantum mechanics. International series in pure and applied physics. University Science Books, 2000.
  13. M. Guenin. On the interaction picture. Commun. Math. Phys., 3:120-132, 1966.
  14. S. Balac and F. Mahe. Embedded Runge-Kutta scheme for step-size control in the Interaction Picture method. Comput. Phys. Commun., 184:1211-1219, 2013.
  15. S. N. Papakostas and G. Papageorgiou. A family of fifth-order RungeKutta pairs. Math. Comp, 65:215, 1996.
  16. J.R. Cash and A.H. Karp. A variable order Runge-Kutta method for initial value problems with rapidly varying right-hand sides". ACM Trans. Math. Software, 16:201-222, 1990.
  17. G. Agrawal. Nonlinear fiber optics. Academic Press, 3rd edition, 2001.
  18. B.M. Caradoc-Davies, R.J. Ballagh, and P.B. Blakie. Three-dimensional vortex dynamics in Bose-Einstein condensates. Phys. Rev. A, 62:011602, 2000.
  19. A. Pazy. Semigroups of Linear Operators and Applications to Partial Differential Equations. Number vol. 44 in Applied Mathematical Sciences. Springer, 1992.
  20. S. Balac, A. Fernandez, F. Mahe, F. Mehats, and R. Texier-Picard. The Interaction Picture method for solving the Generalized Nonlinear Schrodinger Equation in optics. submitted to SIAM J. Numer. Anal., 2013.
  21. G. Strang. On the construction and comparison of difference schemes. SIAM J. Numer. Anal., 5(3):506-517, 1968.
  22. J.C. Butcher. On Runge-Kutta processes of high order. J. Aust. Math. Soc., 4(02):179-194, 1964.
  23. E. Fehlberg. Low order classical Runge-Kutta formulas with stepsize control and applications to some heat transfert problems. Technical report, National Aeronautics and Space Administration, 1969.
  24. J.R. Dormand and P.J. Prince. A family of embedded Runge-Kutta formulae. J. Comput. Appl. Math., 6:19-26, 1980.
  25. W. Kutta. Beitrag zur naherungsweisen integration totaler differentialgleichungen. Z. Math. Phys., (46):434-453, 1901.
  26. C.R. Cassity. The complete solution of the fifth order Runge-Kutta equations. SIAM J. Numer. Anal., 6(3):432-436, 1969.
  27. Maplesoft. Maple 16 Programming Guide. Waterloo Maple Inc., 2012.
  28. L. Shampine. Some practical Runge-Kutta formulas. Math. Comp., 46:135-150, 1986.
  29. J.H.E. Cartwright and O. Piro. The dynamics of Runge-Kutta methods. Int. J. Bifurcation and Chaos, 2:427-49, 1992.
  30. J.D. Lawson. An order five Runge-Kutta process with extended region of stability. SIAM J. Numer. Anal., 3(4):593-597, 1966.
  31. L. Shampine. Local error estimation by doubling. Computing, 34:179-190, 1985.