The secret of The secret of symplectic symplectic

integrators from integrators from the Da Vinci codethe Da Vinci code

Castellón Conference on Geometric Integration 2006

The fundamental The fundamental physicsphysics and and structurestructure

of symplectic of symplectic integratorsintegrators

Siu A. ChinDept of Physics

Texas A&M University

Major themes :Major themes :

1)The physics of the problem 1)The physics of the problem determines the optimal determines the optimal algorithm for its solution.algorithm for its solution.

2)Forward algorithms (all positive 2)Forward algorithms (all positive intermediate time-steps) are the intermediate time-steps) are the “backbones” of composed “backbones” of composed symplectic integrators.symplectic integrators.

Error Hamiltonians cause “perihelion perihelion advancesadvances”.

How 1) worksHow 1) worksin the case of the Kepler orbitin the case of the Kepler orbit

Energy error is periodic.

Precession error accumulates without bound. (Kinoshita et al 91, Gladman et al 91)

Energy error: periodic

Errors Errors in the Kepler problemin the Kepler problem

Precession error:accumulates (LRL vector)

Error coefficients – depend on the mathematical Error coefficients – depend on the mathematical structure of the algorithm.structure of the algorithm.

Error Hamiltonians (Poisson Brackets) – depend Error Hamiltonians (Poisson Brackets) – depend on the physics of the original Hamiltonian.on the physics of the original Hamiltonian.

Error Coefficients and Error HamiltoniansError Coefficients and Error Hamiltonians

Error Hamiltonians of the Kepler Error Hamiltonians of the Kepler Problem – up to the fourth orderProblem – up to the fourth order

Perturbative effect (precession) of each error Perturbative effect (precession) of each error Hamiltonian on the exact orbit can be analytically Hamiltonian on the exact orbit can be analytically computed.computed.

Precession angle per period due to Precession angle per period due to each error Hamiltonianeach error Hamiltonian

1) Each paired error Hamiltonians cause the orbit to precess 1) Each paired error Hamiltonians cause the orbit to precess

oppositely by exactly the same amount after each period!oppositely by exactly the same amount after each period!

2) Precession errors return to zero if paired error coefficients are 2) Precession errors return to zero if paired error coefficients are

equal => equal => corrector/process algorithms corrector/process algorithms ( Wisdom(96), Lopez-Marcos, ( Wisdom(96), Lopez-Marcos,

Sanz-Serna&Skeel(97,97), Blanes,Casas&Ros(99) )Sanz-Serna&Skeel(97,97), Blanes,Casas&Ros(99) )

3) Physics of precession dictates the class of optimal algorithms.3) Physics of precession dictates the class of optimal algorithms.

Second order corrector/process Second order corrector/process algorithms: algorithms: eeTTV TTV = e= eVTVVTV

Velocity-Verlet(VV) – 1 force

Takahash-Imada(TI) – 2 forces

Non-forward – 3 forces

Effectively 4th order

Fourth-order corrector/process algorithmsFourth-order corrector/process algorithms eeTTTTV TTTTV = e= eVTTTV VTTTV , , eeTTVTV TTVTV = e= eVTVTVVTVTV

C’- 6 forces; 4S ~7-8 forces; Effectively 6th orderC - 4 force; Blanes-Moan (BM) 6 forces

Conclusions thus far:Conclusions thus far:

1)1) The Physics of the error The Physics of the error Hamiltonians dictates the optimal Hamiltonians dictates the optimal form of the algorithm.form of the algorithm.

2)2) Most efficient algorithms are those Most efficient algorithms are those tailored to the problem one seeks tailored to the problem one seeks to solve.to solve.

3)3) The age of customized algorithms: The age of customized algorithms: need to know the effects of error need to know the effects of error Hamiltonians (numerically).Hamiltonians (numerically).

The structure of factorized SIThe structure of factorized SIa fundamental theorema fundamental theorem:: (Chin 06)(Chin 06)

ForFor

The errorThe errorcoefficientscoefficientsobeyobey

wherewhere

Implications ofImplications of

2) 2) The first three error coefficients The first three error coefficients cannot all vanish cannot all vanish => Sheng(89)-Suzuki(91)Theorem => Sheng(89)-Suzuki(91)Theorem

1)1) There’s a fundamental and There’s a fundamental and preciseprecise relationship satisfied by relationship satisfied by the first three error coefficients.the first three error coefficients.

Implications ofImplications of

3) Second order corrector/process 3) Second order corrector/process kernel algorithms requirekernel algorithms require

=>No forward corrector kernel algorithms =>No forward corrector kernel algorithms with only T and V operators are possible with only T and V operators are possible beyond first order. (beyond first order. (Chin 04, Blanes & Casas 05)Chin 04, Blanes & Casas 05)

Implications ofImplications of

4) If4) If and are zero, thenand are zero, then

can vanish only if can vanish only if => at least one t=> at least one ti i must be negativemust be negative

=> Goodman-Kaper (96), beyond second=> Goodman-Kaper (96), beyond second order, one torder, one ti i and one vand one vi i must be negative.must be negative.

Implications ofImplications of

5) If5) If and are zero, then and are zero, then fourth-order forward algorithms withfourth-order forward algorithms with

arbitrary number of operators can be arbitrary number of operators can be derived by saturating the inequality, derived by saturating the inequality, by settingby setting

wherewhere

Keeping the Keeping the eeVTVVTV error term means error term means

include the error commutator [V,[T,V]] include the error commutator [V,[T,V]] in factorization schemes in factorization schemes =>=> Forward time-step algorithms Forward time-step algorithms

Quantum case:

Classically: (Chin 97,Ruth 83)and can be evaluated as

Fourth-order forward Fourth-order forward symplectic algorithmssymplectic algorithms

Algorithm C is the symmetrization ofRuth’s 1983 third-order algorithm

Suzuki 96, Chin 97

Chin 97where

Solving time-irreversible problems: Solving time-irreversible problems: Fourth-order Langevin algorithmFourth-order Langevin algorithm

Forbert & Chin 2001. 121 particles Brownian particles in 2D with Yukawa interaction.

Solving time-irreversible problems: Solving time-irreversible problems: Fourth-order Diffusion MC algorithmFourth-order Diffusion MC algorithm

Forbert & Chin 2001.Ground state energy of superfluid Helium at zero K.

Solving time-irreversible problems: Solving time-irreversible problems:

Forbert & Chin 2001.Solid square, circle, others’ algorithm. Rest are variants of forward fourth-order algorithms.

Kramer’s quationKramer’s quation

Solving time-irreversible problems: Solving time-irreversible problems: The rotating Gross-Pitaveski equationThe rotating Gross-Pitaveski equation

Chin & Krotscheck 2005.Chemical potential of a rotating Bose-Einstein Condensate.

Solving time-reversible problems: Solving time-reversible problems: The radial Schrodinger equationThe radial Schrodinger equation

Chin & Anisimov 2006.Ground state of Hydrogen atom in atomic units

Solving time-reversible problems:Solving time-reversible problems: The time-dependent Schrodinger Eq.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.

Solving time-reversible problems:Solving time-reversible problems:A A restricted 3-body problemrestricted 3-body problem

Chin & Chen 2005.

Third body’s orbit in 2D, Chin and Chen 05

Secrets from the Da Vinci Code :Secrets from the Da Vinci Code :

1)1) The general structure of symplectic The general structure of symplectic integrators can be best understood integrators can be best understood by considering forward time-step by considering forward time-step algorithms as primary.algorithms as primary.

2)2) Underlying physics determines the Underlying physics determines the optimal algorithm.optimal algorithm.

3)3) Forward algorithms best all known Forward algorithms best all known algorithms of the same order with algorithms of the same order with the same effort in all cases tested.the same effort in all cases tested.