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The secret of symplectic integrators from the Da Vinci code

The secret of symplectic integrators from the Da Vinci code. Castell ó n Conference on Geometric Integration 2006. The fundamental physics and structure of symplectic integrators. Siu A. Chin Dept of Physics Texas A&M University.

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The secret of symplectic integrators from the Da Vinci code

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  1. The secret of symplectic integrators from the Da Vinci code Castellón Conference on Geometric Integration2006

  2. The fundamental physics and structureof symplectic integrators Siu A. Chin Dept of Physics Texas A&M University

  3. 1)The physics of the problem determines the optimal algorithm for its solution. Major themes : 2)Forward algorithms (all positive intermediate time-steps) are the “backbones” of composed symplectic integrators.

  4. How 1) worksin the case of the Kepler orbit Energy error is periodic. Precession error accumulates without bound. (Kinoshita et al 91, Gladman et al 91) Error Hamiltonians cause “perihelion advances”.

  5. Errors in the Kepler problem Energy error: periodic Precession error: accumulates (LRL vector)

  6. Error Coefficients and Error Hamiltonians Error coefficients – depend on the mathematical structure of the algorithm. Error Hamiltonians (Poisson Brackets) – depend on the physics of the original Hamiltonian.

  7. Error Hamiltonians of the Kepler Problem – up to the fourth order Perturbative effect (precession) of each error Hamiltonian on the exact orbit can be analytically computed.

  8. Precession angle per period due to each error Hamiltonian 1) Each paired error Hamiltonians cause the orbit to precess oppositely by exactly the same amount after each period! 2) Precession errors return to zero if paired error coefficients are equal => corrector/process algorithms ( Wisdom(96), Lopez-Marcos, Sanz-Serna&Skeel(97,97), Blanes,Casas&Ros(99) ) 3) Physics of precession dictates the class of optimal algorithms.

  9. Velocity-Verlet (VV) – 1 force Takahash-Imada (TI) – 2 forces Non-forward – 3 forces Effectively 4th order Second order corrector/process algorithms: eTTV = eVTV

  10. Fourth-order corrector/process algorithmseTTTTV = eVTTTV , eTTVTV = eVTVTV C’- 6 forces; 4S ~7-8 forces; Effectively 6th order C - 4 force; Blanes-Moan (BM) 6 forces

  11. The Physics of the error Hamiltonians dictates the optimal form of the algorithm. • Most efficient algorithms are those tailored to the problem one seeks to solve. • The age of customized algorithms: need to know the effects of error Hamiltonians (numerically). Conclusions thus far:

  12. For The structure of factorized SIa fundamental theorem:(Chin 06) The error coefficients obey where

  13. There’s a fundamental and precise relationship satisfied by the first three error coefficients. Implications of 2) The first three error coefficients cannot all vanish => Sheng(89)-Suzuki(91)Theorem

  14. 3) Second order corrector/process kernel algorithms require Implications of =>No forward corrector kernel algorithms with only T and V operators are possible beyond first order. (Chin 04, Blanes & Casas 05)

  15. 4) Ifand are zero, then Implications of can vanish only if => at least one ti must be negative => Goodman-Kaper (96), beyond second order, one ti and one vi must be negative.

  16. 5) Ifand are zero, then fourth-order forward algorithms with arbitrary number of operators can be derived by saturating the inequality, Implications of by setting where

  17. Keeping the eVTVerror term means include the error commutator [V,[T,V]] in factorization schemes => Forward time-step algorithms Quantum case: Classically: (Chin 97,Ruth 83) and can be evaluated as

  18. Suzuki 96, Chin 97 Fourth-order forward symplectic algorithms Chin 97 where Algorithm C is the symmetrization of Ruth’s 1983 third-order algorithm

  19. Solving time-irreversible problems: Fourth-order Langevin algorithm Forbert & Chin 2001. 121 particles Brownian particles in 2D with Yukawa interaction.

  20. Solving time-irreversible problems: Fourth-order Diffusion MC algorithm Forbert & Chin 2001. Ground state energy of superfluid Helium at zero K.

  21. Kramer’s quation Solving time-irreversible problems: Forbert & Chin 2001. Solid square, circle, others’ algorithm. Rest are variants of forward fourth-order algorithms.

  22. Solving time-irreversible problems: The rotating Gross-Pitaveski equation Chin & Krotscheck 2005. Chemical potential of a rotating Bose-Einstein Condensate.

  23. Solving time-reversible problems: The radial Schrodinger equation Chin & Anisimov 2006. Ground state of Hydrogen atom in atomic units

  24. Solving time-reversible problems:The time-dependent Schrodinger Eq. Chin & Chen 2002. Solving the TDSE with a TD potential: Preston-Walker model of an atom in a strong laser field.

  25. Solving time-reversible problems:A restricted 3-body problem Chin & Chen 2005. Third body’s orbit in 2D, Chin and Chen 05

  26. The general structure of symplectic integrators can be best understood by considering forward time-step algorithms as primary. • Underlying physics determines the optimal algorithm. • Forward algorithms best all known algorithms of the same order with the same effort in all cases tested. Secrets from the Da Vinci Code :

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