The secret of symplectic integrators from the Da Vinci code

1
The secret of symplectic integrators from the Da
Vinci code

• Castellón Conference on Geometric Integration
  2006

2
The fundamental physics and structure of
symplectic integrators

• Siu A. Chin
• Dept of Physics
• Texas AM University

3
Major themes
1)The physics of the problem determines the
optimal algorithm for its solution.
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

• Error Hamiltonians cause perihelion advances.

Energy error is periodic. Precession error
accumulates without bound. (Kinoshita et al 91,
Gladman et al 91)
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 gt
corrector/process algorithms ( Wisdom(96),
Lopez-Marcos, Sanz-SernaSkeel(97,97),
Blanes,CasasRos(99) ) 3) Physics of precession
dictates the class of optimal algorithms.
9
Second order corrector/process algorithms eTTV
eVTV
Velocity-Verlet (VV) 1 force Takahash-Imada (TI
) 2 forces Non-forward 3 forces
Effectively 4th order
10
Fourth-order corrector/process algorithms
eTTTTV eVTTTV , eTTVTV eVTVTV

• C- 6 forces 4S 7-8 forces Effectively 6th
  order
• C - 4 force Blanes-Moan (BM) 6 forces

11
Conclusions thus far

1. The Physics of the error Hamiltonians dictates
   the optimal form of the algorithm.
2. Most efficient algorithms are those tailored to
   the problem one seeks to solve.
3. The age of customized algorithms need to know
   the effects of error Hamiltonians (numerically).

12
The structure of factorized SIa fundamental
theorem (Chin 06)
For
The error coefficients obey
where
13
Implications of

1. Theres a fundamental and precise relationship
   satisfied by the first three error coefficients.

2) The first three error coefficients cannot all
vanish gt Sheng(89)-Suzuki(91)Theorem
14
Implications of

3) Second order corrector/process kernel
algorithms require

gtNo forward corrector kernel algorithms with
only T and V operators are possible beyond first
order. (Chin 04, Blanes Casas 05)
15
Implications of
4) If and are zero, then

can vanish only if
gt at least one ti must be negative gt
Goodman-Kaper (96), beyond second order,
one ti and one vi must be negative.

16
Implications of
5) If and are zero, then

fourth-order forward algorithms
with arbitrary number of operators
can be derived by saturating the
inequality,
by setting
where
17
Keeping the eVTV error term means include the
error commutator V,T,V in factorization
schemes gt Forward time-step algorithms

• Quantum case
• Classically (Chin
  97,Ruth 83)

and can be evaluated as

18
Fourth-order forward symplectic algorithms
Suzuki 96, Chin 97
Chin 97
where

• Algorithm C is the symmetrization of
• Ruths 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
Solving time-irreversible problems
Kramers quation

• 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 problemsA restricted
3-body problem

• Chin Chen 2005.

Third bodys orbit in 2D, Chin and Chen 05
26
Secrets from the Da Vinci Code

1. The general structure of symplectic integrators
   can be best understood by considering forward
   time-step algorithms as primary.
2. Underlying physics determines the optimal
   algorithm.
3. Forward algorithms best all known algorithms of
   the same order with the same effort in all cases
   tested.