Advertisement

Finite difference approximation for nonlinear Schrödinger equations with application to blow-up computation

  • Norikazu SaitoEmail author
  • Takiko Sasaki
Original Paper Area 1

Abstract

This paper presents a coherent analysis of the finite difference method to nonlinear Schrödinger (NLS) equations in one spatial dimension. We use the discrete \(H^1\) framework to establish well-posedness and error estimates in the \(L^\infty \) norm. The nonlinearity f(u) of a NLS equation is assumed to satisfy only a growth condition. We apply our results to computation of blow-up solutions for a NLS equation with the nonlinearity \(f(u)=-|u|^{2p}\), p being a positive real number. Particularly, we offer the numerical blow-up time \(T(h,\tau )\), where h and \(\tau \) are discretization parameters of space and time variables. We prove that \(T(h,\tau )\) converges to the blow-up time \(T_\infty \) of the solution of the original NLS equation. Several numerical examples are presented to confirm the validity of theoretical results. Furthermore, we infer from numerical investigation that the convergence of \(T(h,\tau )\) is at a second order rate in \(\tau \) if the Crank–Nicolson scheme is applied to time discretization.

Keywords

Nonlinear Schrödinger equation Blow-up Finite difference method 

Mathematics Subject Classification

35Q55 35B44 65M06 

Notes

Acknowledgments

This work was supported by CREST, Japan Science and Technology Agency, and JSPS KAKENHI Grant Numbers 15H03635 and 15K13454.

References

  1. 1.
    Akrivis, G.D., Dougalis, V.A., Karakashian, O.A.: On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation. Numer. Math. 59, 31–53 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Akrivis, G.D., Dougalis, V.A., Karakashian, O.A., McKinney, W.R.: Numerical approximation of blow-up of radially symmetric solutions of the nonlinear Schr\(\ddot{o}\)dinger equation. SIAM J. Sci. Comput. 25, 186–212 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bao, W., Cai, Y.: Uniform error estimates of finite difference methods for the nonlinear Schrödinger equation with wave operator. SIAM J. Numer. Anal. 50, 492–521 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Besse, C., Carles, R., Mauser, N.J., Stimming, H.P.: Monotonicity properties of the blow-up time for nonlinear Schrödinger equations, numerical evidence. Discrete Contin. Dyn. Syst. Ser. B 9, 11–36 (2008)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Cazenave, T.: Semilinear Schrödinger Equations. AMS, New York (2003)zbMATHGoogle Scholar
  6. 6.
    Chen, Y.G.: Asymptotic behaviours of blowing-up solutions for finite difference analogue of \(u_t=u_{xx}+u^{1+\alpha }\). J. Fac. Sci. Univ. Tokyo Sect. IA Math. 33, 541–574 (1986)MathSciNetGoogle Scholar
  7. 7.
    Cho, C.H.: A finite difference scheme for blow-up solutions of nonlinear wave equations. Numer. Math. Theory Methods Appl. 3, 475–498 (2010)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Cho, C.H., Hamada, S., Okamoto, H.: On the finite difference approximation for a parabolic blow-up problem. Japan J. Indust. Appl. Math. 24, 131–160 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chang, Q., Jia, E., Sun, W.: Difference schemes for solving the generalized nonlinear Schrödinger equation. J. Comput. Phys. 148, 397–415 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Heywood, J.G., Rannacher, R.: Finite-element approximation of the nonstationary Navier–Stokes problem. IV. Error analysis for second-order time discretization. SIAM J. Numer. Anal. 27, 353–384 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kato, T.: On nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Phys. Théor. 46, 113–129 (1987)zbMATHGoogle Scholar
  12. 12.
    Kavian, O.: A remark on the blowing-up of solutions to the Cauchy problem for nonlinear Schrödinger equations. Trans. Am. Math. Soc. 299, 193–203 (1987)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Merle, F., Raphael, P.: On universality of blow-up profile for \(L^2\) critical nonlinear Schrödinger equation. Invent. Math. 156, 565–672 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Nakagawa, T.: Blowing up of a finite difference solution to \(u_t=u_{xx}+u^2\). Appl. Math. Optim. 2, 337–350 (1975/76)Google Scholar
  15. 15.
    Nakagawa, T., Ushijima, T.: Finite element analysis of the semi-linear heat equation of blow-up type. In: Topics Numer. Anal. III. Academic Press, New York (1977)Google Scholar
  16. 16.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin (1992)zbMATHGoogle Scholar
  17. 17.
    Saito, N., Sasaki, T.: Blow-up of finite-difference solutions to nonlinear wave equations. J. Math. Sci. Univ. Tokyo, 23(1), 349–380 (2016)Google Scholar
  18. 18.
    Segal, I.: Non-linear semi-groups. Ann. Math. 78, 339–364 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Sun, Z., Zhao, D.: On the \(L^\infty \) convergence of a difference scheme for coupled nonlinear Schrödinger equations. Comput. Math. Appl. 59, 3286–3300 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Thomée, V.: Finite difference methods for linear parabolic equations. In: Handbook of Numerical Analysis, vol. I. North-Holland, The Netherlands, pp. 5–196 (1990)Google Scholar
  21. 21.
    Ushijima, T.K.: On the approximation of blow-up time for solutions of nonlinear parabolic equations. Publ. Res. Inst. Math. Sci. 36, 613–640 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Wang, J.: A new error analysis of Crank–Nicolson Galerkin FEMs for a generalized nonlinear Schrödinger equation. J. Sci. Comput. 60, 390–407 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Zhang, J.: On the finite-time behaviour for nonlinear Schrödinger equations. Commun. Math. Phys. 162, 249–260 (1994)CrossRefzbMATHGoogle Scholar
  24. 24.
    Zhou, G., Saito, N.: Finite volume methods for a Keller–Segel system: discrete energy, error estimates and numerical blow-up analysis. Numer. Math. doi: 10.1007/s00211-016-0793-2

Copyright information

© The JJIAM Publishing Committee and Springer Japan 2016

Authors and Affiliations

  1. 1.Graduate School of Mathematical SciencesThe University of TokyoTokyoJapan
  2. 2.Global Education CenterWaseda UniversityTokyoJapan

Personalised recommendations