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An Accurate Approximation of Exponential Integrators for the Schrödinger Equation

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Abstract

Numerical time propagation of semi-linear equations of the Schrödinger type can be performed by the use of exponential integrators. The main difficulty for efficient implementation of this type of schemes lies in the evaluation of \(\varphi \)-functions of a matrix argument. We develop a Chebyshev series approximation for these functions and propose a simple algorithm for the evaluation of the series coefficients. The domain of convergence of the series is consistent with the spectrum of Schrödinger type operators. This approximation is shown to be accurate and performs favorably in comparison to other state of the art methods for approximation of \(\varphi \)-functions.

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References

  1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1972)

    MATH  Google Scholar 

  2. Bao, W., Jaksch, D.: An explicit unconditionally stable numerical method for solving damped nonlinear Schrödinger equations with a focusing nonlinearity. SIAM J. Numer. Anal. 41, 1406–1426 (2003)

    MathSciNet  MATH  Google Scholar 

  3. Besse, C.: A relaxation scheme for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 42, 934–952 (2004)

    MathSciNet  MATH  Google Scholar 

  4. Besse, C., Bidégaray, B., Descombes, S.: Order estimates in time of splitting methods for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 40, 26–40 (2002)

    MathSciNet  MATH  Google Scholar 

  5. Besse, C., Dujardin, G., Lacroix-Violet, I.: High order exponential integrators for nonlinear Schrödinger equations with application to rotating Bose–Einstein condensates. SIAM J. Numer. Anal. 55, 1387–1411 (2017)

    MathSciNet  MATH  Google Scholar 

  6. Beylkin, G., Keiser, J.M., Vozovoi, L.: A new class of time discretization schemes for the solution of nonlinear PDEs. J. Comput. Phys. 147, 362–387 (1998)

    MathSciNet  MATH  Google Scholar 

  7. Caliari, M.: Accurate evaluation of divided differences for polynomial interpolation of exponential propagators. Computing 80, 189–201 (2007)

    MathSciNet  MATH  Google Scholar 

  8. Caliari, M., Ostermann, A.: Implementation of exponential Rosenbrock-type integrators. Appl. Numer. Math. 59, 568–581 (2009)

    MathSciNet  MATH  Google Scholar 

  9. Calvo, M.P., Palencia, C.: A class of explicit multistep exponential integrators for semilinear problems. Numer. Math. 102, 367–381 (2006)

    MathSciNet  MATH  Google Scholar 

  10. Calvo, M.P., Portillo, A.M.: Variable step implementation of ETD methods for semilinear problems. Appl. Math. Comput. 196, 627–637 (2008)

    MathSciNet  MATH  Google Scholar 

  11. Cohen, D., Gauckler, L.: One-stage exponential integrators for nonlinear Schrödinger equations over long times. BIT 52, 877–903 (2012)

    MathSciNet  MATH  Google Scholar 

  12. Chan, T.F., Lee, D., Shen, L.: Stable explicit schemes for equations of the Schrödinger type. SIAM J. Numer. Anal. 23, 274–281 (1986)

    MathSciNet  MATH  Google Scholar 

  13. Celledoni, E., Cohen, D., Owren, B.: Symmetric exponential integrators with an application to the cubic Schrödinger equation. Found. Comput. Math. 8, 303–317 (2008)

    MathSciNet  MATH  Google Scholar 

  14. Cox, S.M., Matthews, P.C.: Exponential time differencing for stiff systems. J. Comput. Phys. 176, 430–455 (2002)

    MathSciNet  MATH  Google Scholar 

  15. Cryer, C.W.: The difference analogue of Gauss’ theorem. SIAM J. Numer. Anal. 4, 155–162 (1967)

    MathSciNet  MATH  Google Scholar 

  16. Fei, Z., Pérez-García, V.M., Vázquez, L.: Numerical simulation of nonlinear Schrödinger systems: a new conservative scheme. Appl. Math. Comput. 71, 165–177 (1995)

    MathSciNet  MATH  Google Scholar 

  17. Göckler, T., Grimm, V.: Uniform approximation of \(\varphi \)-functions in exponential integrators by a rational Krylov subspace method with simple poles. SIAM J. Matrix Anal. Appl. 35, 1467–1489 (2014)

    MathSciNet  MATH  Google Scholar 

  18. Higham, N.J.: Accuracy and Stability of Numerical Algorithms. SIAM, Philadelphia (1996)

    MATH  Google Scholar 

  19. Hochbruck, M., Lubich, Ch., Selhofer, H.: Exponential integrators for large systems of differential equations. SIAM J. Sci. Comput. 19, 1552–1574 (1998)

    MathSciNet  MATH  Google Scholar 

  20. Hochbruck, M., Ostermann, A., Schweitzer, J.: Exponential Rosenbrock-type methods. SIAM J. Numer. Anal. 47, 786–803 (2009)

    MathSciNet  MATH  Google Scholar 

  21. Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numer. 19, 209–286 (2010)

    MathSciNet  MATH  Google Scholar 

  22. Kassam, A.-K., Trefethen, L.N.: Fourth-order time-stepping for stiff PDEs. SIAM J. Sci. Comput. 26, 1214–1233 (2005)

    MathSciNet  MATH  Google Scholar 

  23. Krogstad, S.: Generalized integrating factor methods for stiff PDEs. J. Comput. Phys. 203, 7288 (2005)

    MathSciNet  MATH  Google Scholar 

  24. Luke, Y.L.: Integrals of Bessel Functions. McGraw-Hill, New York (1962)

    MATH  Google Scholar 

  25. Maia, L.A., Montefusco, E., Pellacci, B.: Positive solutions for a weakly coupled nonlinear Schrödinger system. J. Differ. Equ. 229, 743–767 (2006)

    MATH  Google Scholar 

  26. Meinardus, G.: Approximation of Functions: Theory and Numerical Methods. Springer, New York (1967)

    MATH  Google Scholar 

  27. Minchev, B., Wright, W.M.: A review of exponential integrators for semilinear problems, Technical Report 2/05. Department of Mathematical Sciences, NTNU, Norway (2005)

  28. Niesen, J., Wright, W.M.: Algorithm 919: a Krylov subspace algorithm for evaluating the \(\varphi \)-functions appearing in exponential integrators. ACM Trans. Math. Softw. 38, 22:1–22:19 (2012)

    MathSciNet  MATH  Google Scholar 

  29. Reichel, L.: Newton interpolation at Leja points. BIT 30, 332–346 (1990)

    MathSciNet  MATH  Google Scholar 

  30. Schaefer, I., Tal-Ezer, H., Kosloff, R.: Semi-global approach for propagation of the time-dependent Schrödinger equation for time-dependent and nonlinear problems. J. Comput. Phys. 343, 368–413 (2017)

    MathSciNet  MATH  Google Scholar 

  31. Schmelzer, T., Trefethen, L.N.: Evaluating matrix functions for exponential integrators via Caratheodory–Fejer approximation and contour integrals. Electron. Trans. Numer. Anal. 29, 1–18 (2007)

    MathSciNet  MATH  Google Scholar 

  32. Skaflestad, B., Wright, W.M.: The scaling and modified squaring method for matrix functions related to the exponential. Appl. Numer. Math. 59, 783–799 (2009)

    MathSciNet  MATH  Google Scholar 

  33. Suhov, A.Y.: A spectral method for the time evolution in parabolic problems. J. Sci. Comput. 29, 201–217 (2006)

    MathSciNet  MATH  Google Scholar 

  34. Suhov, A.Y.: An accurate polynomial approximation of exponential integrators. J. Sci. Comput. 60, 684–698 (2014)

    MathSciNet  MATH  Google Scholar 

  35. Tal-Ezer, H.: On restart and error estimation for Krylov approximation of \(w = f(A)v\). SIAM J. Sci. Comput. 29, 2426–2441 (2007)

    MathSciNet  MATH  Google Scholar 

  36. Tal-Ezer, H., Kosloff, R.: An accurate and efficient scheme for propagating the time dependent Schrödinger equation. J. Chem. Phys. 81, 3967–3971 (1984)

    Google Scholar 

  37. Toh, K.-C., Trefethen, L.N.: The Kreiss matrix theorem on a general complex domain. SIAM J. Matrix Anal. Appl. 21, 145–165 (1999)

    MathSciNet  MATH  Google Scholar 

  38. Tokman, M.: Efficient integration of large stiff systems of ODEs with exponential propagation iterative (EPI) methods. J. Comput. Phys. 213(2), 748–776 (2006)

    MathSciNet  MATH  Google Scholar 

  39. Tokman, M., Loffeld, J., Tranquilli, P.: New adaptive exponential propagation iterative methods of Runge–Kutta type. SIAM J. Sci. Comput. 34, A2650–A2669 (2012)

    MathSciNet  MATH  Google Scholar 

  40. Rainwater, G., Tokman, M.: A new class of split exponential propagation iterative methods of Runge–Kutta type (sEPIRK) for semilinear systems of ODEs. J. Comput. Phys. 269, 40–60 (2014)

    MathSciNet  MATH  Google Scholar 

  41. Trefethen, L.N.: Spectral Methods in Matlab. SIAM, Philadelphia (2000)

    MATH  Google Scholar 

  42. Trefethen, L.N.: Is Gauss quadrature better than Clenshaw–Curtis? SIAM Rev. 50(1), 67–87 (2008)

    MathSciNet  MATH  Google Scholar 

  43. Wu, L.: Dufort-Frankel-type methods for linear and nonlinear Schrödinger. SIAM J. Numer. Anal. 33, 1526–1533 (1996)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author wishes to thank Adi Ditkowski for valuable discussions of the method presented in this paper.

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  1. Department of Condensed Matter Physics, Weizmann Institute of Science, 76100, Rehovot, Israel

    A. Y. Meltzer

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  1. A. Y. Meltzer
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Correspondence to A. Y. Meltzer.

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Meltzer, A.Y. An Accurate Approximation of Exponential Integrators for the Schrödinger Equation. J Sci Comput 81, 1493–1508 (2019). https://doi.org/10.1007/s10915-019-01075-1

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