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Sergio Blanes Joint work with Fernando Casas and Ander Murua

Splitting Methods for the time-dependent Schödinger equation. Sergio Blanes Joint work with Fernando Casas and Ander Murua Instituto de Matemática Multidisciplinar Universidad Politécnica de Valencia, SPAIN Workshop on Splitting Methods in Time Integration

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Sergio Blanes Joint work with Fernando Casas and Ander Murua

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  1. Splitting Methods for the time-dependent Schödinger equation Sergio Blanes Joint work with Fernando Casas and Ander Murua Instituto de Matemática Multidisciplinar Universidad Politécnica de Valencia, SPAIN Workshop on Splitting Methods in Time Integration Innsbruck, 15-18 October 2008

  2. Consider the linear time dependent SE It is separable in its kinetic and potential parts. The solution of the discretised equation is given by where c = (c1,…, cN)T CN and H = T + V RN×N Hermitian matrix. Fourier methods are frequently used is the fast Fourier transform (FFT)

  3. Consider the Strang-splitting or leap-frog second order method Notice that the exponentials are computed only once and are stored at the beginning. Similarly, for the kinetic part we have

  4. Consider then Hamiltonian system: with formal solution It is an orthogonal and symplectic operator.

  5. Consider then Hamiltonian system: with formal solution It is an orthogonal and symplectic operator. Takingz=(q,p), then z’=(A+B)z Notice that

  6. Efficiency of a Method versus ErrorComputational Cost

  7. Higher Orders The well known fourth-order composition scheme with

  8. Higher Orders The well known fourth-order composition scheme with Or in general with -Creutz-Gocksh(89) -Suzuki(90) -Yoshida(90)

  9. Higher Orders The well known fourth-order composition scheme with Or in general with -Creutz-Gocksh(89) Excellent trick!! -Suzuki(90) -Yoshida(90) In practice, they show low performance!!

  10. Order 4 6 8 10 3- For-Ru(89) Yos(90),ect. 5-Suz(90) McL(95)

  11. Order 4 6 8 10 3- For-Ru(89) 7-Yoshida(90) Yos(90),etc. 9-McL(95) 5-Suz(90) Kahan-Li(97) McL(95) 11-13-Sof-Spa(05)

  12. Order 4 6 8 10 3- For-Ru(89) 7-Yoshida(90) 15-Yoshida(90) Yos(90),etc. 9-McL(95) Suz-Um(93) 5-Suz(90) Kahan-Li(97) 15-17-McL(95) McL(95) 11-13-Sof-Spa(05) Kahan-Li(97) 19-21-Sof-Spa(05) 24SS-Ca-SS(93)

  13. Order 4 6 8 10 3- For-Ru(89) 7-Yoshida(90) 15-Yoshida(90) 31-Suz-Um(93) Yos(90),etc. 9-McL(95) Suz-Um(93) 31-33-Ka-Li(97) 5-Suz(90) Kahan-Li(97) 15-17-McL(95) 33-Tsitouras(00) McL(95) 11-13-Sof-Spa(05) Kahan-Li(97) 31-35-Wanner(02) 19-21-Sof-Spa(05) 31-35-Sof-Spa(05) 24SS-Ca-SS(93)

  14. Order 4 6 8 10 3- For-Ru(89) 7-Yoshida(90) 15-Yoshida(90) 31-Suz-Um(93) Yos(90),etc. 9-McL(95) Suz-Um(93) 31-33-Ka-Li(97) 5-Suz(90) Kahan-Li(97) 15-17-McL(95) 33-Tsitouras(00) McL(95) 11-13-Sof-Spa(05) Kahan-Li(97) 31-35-Wanner(02) 19-21-Sof-Spa(05) 31-35-Sof-Spa(05) 24SS-Ca-SS(93) Processed P3-17-McL(02) P5-15 P9-19 P15-25 B-Casas-Murua(06)

  15. Order 4 6 8 10 12 4,668 10 12 Gray-Manolopoulos(96) 11 11 17 B-Casas-Murua(08) Processed P3,4 P3,4 P4-5 McL-Gray(97) P2-40orders: 2-20 B-Casas-Murua(06)

  16. Let us consider again the linear time dependent SE We have the discrete form Consider then Hamiltonian system: with formal solution

  17. We have built splitting methods for the harmonic oscillator!!! Exact solution (ortogonal and symplectic) We consider the composition Notice that

  18. Example: Harmonic oscillator 100 periods with initial conditions (q,p)=(1,1)

  19. Example: Harmonic oscillator 100 periods with initial conditions (q,p)=(1,1)

  20. Example: Harmonic oscillator 100 periods with initial conditions (q,p)=(1,1)

  21. Example: Harmonic oscillator 100 periods with initial conditions (q,p)=(1,1)

  22. Example: Harmonic oscillator 100 periods with initial conditions (q,p)=(1,1)

  23. Example: Harmonic oscillator 10000 periods with initial conditions (q,p)=(1,1)

  24. Example: Harmonic oscillator Error growth in (q,p) along 10000 periods

  25. Schrödinger equation with a Morse potential with

  26. Schrödinger equation with a Morse potential with Initial conditions

  27. Gaussian wave fuction in a Morse potential

  28. Gaussian wave fuction in a Morse potential

  29. Gaussian wave fuction in a Morse potential

  30. Gaussian wave fuction in a Morse potential

  31. Applications to other Problems Maxwell Equations electric & magnetic field vectors permeability and permitivity We consider dimensionless equations (by choosing the units properly). With a faithful spatial representation of E and B we have the discrete form

  32. Applications to other Problems To numerically solve the linear time dependent system and the solution can not be written in a closed form The simplest solution is to convert the system into autonomous where The system is no longer linear and the most efficient methods can not be used We consider appropriate time averages to approximate the solution up to a given order in the time step. This allows us to use the previous techniques developed for splitting methods

  33. Example: Harmonic oscillator 100 periods with initial conditions (q,p)=(1,1)

  34. Applications to other Problems To numerically solve the linear time dependent system and the solution can not be written in a closed form The simplest solution is to convert the system into autonomous where The system is no longer linear and the most efficient methods can not be used We consider appropriate time averages to approximate the solution up to a given order in the time step. This allows us to use the previous techniques developed for splitting methods

  35. Let us consider then with We approximate the solution by the composition with the averages

  36. Gaussian wave fuction in a Morse potential with laser field

  37. Conclusions • Splitting methods are powerful tools for numerically solving many problems

  38. Conclusions • Splitting methods are powerful tools for numerically solving many problems • But, the performance strongly dependes on the choice of appropriate methods for each problem

  39. Conclusions • Splitting methods are powerful tools for numerically solving many problems • But, the performance strongly dependes on the choice of appropriate methods for each problem • Alternatively, one can build methods tailored for particular problems

  40. Splitting Methods for the time-dependent Schödinger equation Sergio Blanes Joint work with Fernando Casas and Ander Murua Instituto de Matemática Multidisciplinar Universidad Politécnica de Valencia, SPAIN Workshop on Splitting Methods in Time Integration Innsbruck, 15-18 October 2008

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