Foundations of Computational Mathematics

, Volume 8, Issue 3, pp 303–317 | Cite as

Symmetric Exponential Integrators with an Application to the Cubic Schrödinger Equation



In this article, we derive and study symmetric exponential integrators. Numerical experiments are performed for the cubic Schrödinger equation and comparisons with classical exponential integrators and other geometric methods are also given. Some of the proposed methods preserve the L2-norm and/or the energy of the system.


Exponential integrators Symmetric methods Cubic Schrödinger equation 

Mathematics Subject Classification (2000)

65P10 35Q55 65P05 


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  1. 1.
    M. J. Ablowitz and J. F. Ladik, A nonlinear difference scheme and inverse scattering, Stud. in Appl. Math. 55(3) (1976), 213–229. MathSciNetGoogle Scholar
  2. 2.
    H. Berland, A. L. Islas, and C. M. Schober, Conservation of phase space properties using exponential integrators on the cubic Schrödinger equation, Technical report 1/06, Norwegian Institute of Science and Technology (2006), submitted to J. Comput. Phys. Google Scholar
  3. 3.
    H. Berland, B. Skaflestad, and W. M. Wright, EXPINT—A MATLAB package for exponential integrators, ACM Trans. Math. Softw. 33(1) (2007). Google Scholar
  4. 4.
    C. Besse, A relaxation scheme for the nonlinear Schrödinger equation, SIAM J. Numer. Anal. 42(3) (2004), 934–952 (electronic). CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    T. J. Bridges and S. Reich, Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity, Phys. Lett. A 284(4/5) (2001), 184–193. CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    B. Cano, Conserved quantities of some Hamiltonian wave equations after full discretization, Numer. Math. 103(2) (2006), 197–223. CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    J. Certaine, The solution of ordinary differential equations with large time constants, in Mathematical Methods for Digital Computers, pp. 128–132, Wiley, New York, 1960. Google Scholar
  8. 8.
    A. Durán and J. M. Sanz-Serna, The numerical integration of relative equilibrium solutions. The nonlinear Schrödinger equation, IMA J. Numer. Anal. 20(2) (2000), 235–261. CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Z. Fei, V. M. Pérez-García, and L. Vázquez, Numerical simulation of nonlinear Schrödinger systems: a new conservative scheme, Appl. Math. Comput. 71(2/3) (1995), 165–177. CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    E. Hairer, Symmetric projection methods for differential equations on manifolds, BIT 40(4) (2000), 726–734. CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edn., Springer Series in Computational Mathematics, Vol. 31, Springer, Berlin, 2006. MATHGoogle Scholar
  12. 12.
    A. L. Islas, D. A. Karpeev, and C. M. Schober, Geometric integrators for the nonlinear Schrödinger equation, J. Comput. Phys. 173(1) (2001), 116–148. CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    A. L. Islas and C. M. Schober, On the preservation of phase space structure under multisymplectic discretization, J. Comput. Phys. 197(2) (2004), 585–609. CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    J. D. Lawson, Generalized Runge–Kutta processes for stable systems with large Lipschitz constants, SIAM J. Numer. Anal. 4 (1967), 372–380. CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, 2nd edn., Texts in Applied Mathematics, Vol. 17, Springer, New York, 1999. MATHGoogle Scholar
  16. 16.
    R. I. McLachlan, Symplectic integration of Hamiltonian wave equations, Numer. Math. 66(4) (1994), 465–492. MathSciNetMATHGoogle Scholar
  17. 17.
    B. Minchev and W. M. Wright, A review of exponential integrators for semilinear problems, Technical report 2/05, Department of Mathematical Sciences, NTNU, Norway (2005),
  18. 18.
    S. Reich, Multi-symplectic Runge–Kutta collocation methods for Hamiltonian wave equations, J. Comput. Phys. 157(2) (2000), 473–499. CrossRefMathSciNetMATHGoogle Scholar
  19. 19.
    T. R. Taha and J. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation, J. Comput. Phys. 55(2) (1984), 203–230. CrossRefMathSciNetMATHGoogle Scholar
  20. 20.
    A. Zanna, K. Engø, and H. Munthe-Kaas, Adjoint and selfadjoint Lie-group methods, BIT 41(2) (2001), 395–421. CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© SFoCM 2007

Authors and Affiliations

  1. 1.Department of Mathematical SciencesNTNUTrondheimNorway

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