Symplectic integrator

IntroductionEdit

Symplectic integrators are designed for the numerical solution of Hamilton's equations, which read

 

where   denotes the position coordinates,   the momentum coordinates, and   is the Hamiltonian. The set of position and momentum coordinates   are called canonical coordinates. (See Hamiltonian mechanics for more background.)

The time evolution of Hamilton's equations is a symplectomorphism, meaning that it conserves the symplectic two-form  . A numerical scheme is a symplectic integrator if it also conserves this two-form.

Symplectic integrators possess, as a conserved quantity, a Hamiltonian which is slightly perturbed from the original one. By virtue of these advantages, the SI scheme has been widely applied to the calculations of long-term evolution of chaotic Hamiltonian systems ranging from the Kepler problem to the classical and semi-classical simulations in molecular dynamics.

Most of the usual numerical methods, like the primitive Euler scheme and the classical Runge–Kutta scheme, are not symplectic integrators.

Methods for constructing symplectic algorithmsEdit

Splitting methods for separable HamiltoniansEdit

A widely used class of symplectic integrators is formed by the splitting methods.

Assume that the Hamiltonian is separable, meaning that it can be written in the form

 

This happens frequently in Hamiltonian mechanics, with T being the kinetic energy and V the potential energy.

For the notational simplicity, let us introduce the symbol   to denote the canonical coordinates including both the position and momentum coordinates. Then, the set of the Hamilton's equations given in the introduction can be expressed in a single expression as

 

where   is a Poisson bracket. Furthermore, by introducing an operator  , which returns a Poisson bracket of the operand with the Hamiltonian, the expression of the Hamilton's equation can be further simplified to

 

The formal solution of this set of equations is given as a matrix exponential:

 

Note the positivity of   in the matrix exponential.

When the Hamiltonian has the form of eq. (1), the solution (3) is equivalent to

 

The SI scheme approximates the time-evolution operator   in the formal solution (4) by a product of operators as

 

where   and   are real numbers,   is an integer, which is called the order of the integrator, and where  . Note that each of the operators   and   provides a symplectic map, so their product appearing in the right-hand side of (5) also constitutes a symplectic map.

Since   for all  , we can conclude that

 

By using a Taylor series,   can be expressed as

 

where   is an arbitrary real number. Combining (6) and (7), and by using the same reasoning for   as we have used for  , we get

 

In concrete terms,   gives the mapping

 

and   gives

 

Note that both of these maps are practically computable.

ExamplesEdit

The simplified form of the equations are:

 
 

After converting into Lagrangian coordinates:

 
 

Where   is the force vector at  , and   is the scalar quantity of mass.

Several symplectic integrators are given below. An illustrative way to use them is to consider a particle with position   and velocity  .

To apply a timestep with values   to the particle, carry out the following steps:

Recursively:

  • Update the position   of the particle by adding to it its (previously updated) velocity   multiplied by  
  • Update the velocity   of the particle by adding to it its acceleration (at updated position) multiplied by  

A first-order exampleEdit

The symplectic Euler method is the first-order integrator with   and coefficients

 

Note that the algorithm above does not work if time-reversibility is needed. The algorithm has to be implemented in two parts, one for positive time steps, one for negative time steps.

A second-order exampleEdit

The Verlet method is the second-order integrator with   and coefficients

 

Since  , the algorithm above is symmetric in time. There are 3 steps to the algorithm, and step 1 and 3 are exactly the same, so the positive time version can be used for negative time.

A third-order exampleEdit

A third-order symplectic integrator (with  ) was discovered by Ronald Ruth in 1983.[1] One of the many solutions is given by

 

A fourth-order exampleEdit

A fourth-order integrator (with  ) was also discovered by Ruth in 1983 and distributed privately to the particle-accelerator community at that time. This was described in a lively review article by Forest.[2] This fourth-order integrator was published in 1990 by Forest and Ruth and also independently discovered by two other groups around that same time.[3][4][5]

 

To determine these coefficients, the Baker–Campbell–Hausdorff formula can be used. Yoshida, in particular, gives an elegant derivation of coefficients for higher-order integrators. Later on, Blanes and Moan[6] further developed partitioned Runge–Kutta methods for the integration of systems with separable Hamiltonians with very small error constants.

Splitting methods for general nonseparable HamiltoniansEdit

General nonseparable Hamiltonians can also be explicitly and symplectically integrated.

To do so, Tao introduced a restraint that binds two copies of phase space together to enable an explicit splitting of such systems.[7] The idea is, instead of  , one simulates  , whose solution agrees with that of   in the sense that  .

The new Hamiltonian is advantageous for explicit symplectic integration, because it can be split into the sum of three sub-Hamiltonians,  ,  , and  . Exact solutions of all three sub-Hamiltonians can be explicitly obtained: both   solutions correspond to shifts of mismatched position and momentum, and   corresponds to a linear transformation. To symplectically simulate the system, one simply composes these solution maps.

ApplicationsEdit

In Plasma PhysicsEdit

In recent decades symplectic integrator in plasma physics has become an active research topic,[8] because straightforward applications of the standard symplectic methods do not suit the need of large-scale plasma simulations enabled by the peta- to exa-scale computing hardware. Special symplectic algorithms need to be customarily designed, tapping into the special structures of physics problem under investigation. One such example is the charged particle dynamics in an electromagnetic field. With the canonical symplectic structure, the Hamiltonian of the dynamics is

 
whose  -dependence and  -dependence are not separable, and standard explicit symplectic methods do not apply. For large-scale simulations on massively parallel clusters, however, explicit methods are preferred. To overcome this difficulty, we can explore the specific way that the  -dependence and  -dependence are entangled in this Hamiltonian, and try to design a symplectic algorithm just for this or this type of problem. First, we note that the  -dependence is quadratic, therefore the first order symplectic Euler method implicit in   is actually explicit. This is what is used in the canonical symplectic particle-in-cell (PIC) algorithm.[9] To build high order explicit methods, we further note that the  -dependence and  -dependence in this   are product-separable, and 2nd and 3rd order explicit symplectic algorithms can be constructed using generating functions.[10]

A more elegant and versatile alternative is to look at the following non-canonical symplectic structure of the problem,

 
Here   is a non-constant non-canonical symplectic form. General symplectic integrator for non-constant non-canonical symplectic structure, explicit or implicit, is not known to exist. However, for this specific problem, a family of high-order explicit non-canonical symplectic integrators can be constructed using the He splitting method.[11] Splitting   into 4 parts,
 
we find serendipitously that for each subsystem, e.g.,
 
and
 
the solution map can be written down explicitly and calculated exactly. Then explicit high-order non-canonical symplectic algorithms can be constructed using different compositions. Let   and   denote the exact solution maps for the 4 subsystems. A 1st-order symplectic scheme is
 
A symmetric 2nd-order symplectic scheme is,
 
which is a customarily modified Strang splitting. A  -th order scheme can be constructed from a  -th order scheme using the method of triple jump,
 
The He splitting method is one of key techniques used in the structure-preserving geometric particle-in-cell (PIC) algorithms [12][13][14] .

See alsoEdit

ReferencesEdit

  1. ^ Ruth, Ronald D. (August 1983). "A Canonical Integration Technique". IEEE Transactions on Nuclear Science. NS-30 (4): 2669–2671. Bibcode:1983ITNS...30.2669R. doi:10.1109/TNS.1983.4332919.
  2. ^ Forest, Etienne (2006). "Geometric Integration for Particle Accelerators". J. Phys. A: Math. Gen. 39 (19): 5321–5377. Bibcode:2006JPhA...39.5321F. doi:10.1088/0305-4470/39/19/S03.
  3. ^ Forest, E.; Ruth, Ronald D. (1990). "Fourth-order symplectic integration" (PDF). Physica D. 43: 105. Bibcode:1990PhyD...43..105F. doi:10.1016/0167-2789(90)90019-L.
  4. ^ Yoshida, H. (1990). "Construction of higher order symplectic integrators". Phys. Lett. A. 150 (5–7): 262. Bibcode:1990PhLA..150..262Y. doi:10.1016/0375-9601(90)90092-3.
  5. ^ Candy, J.; Rozmus, W (1991). "A Symplectic Integration Algorithm for Separable Hamiltonian Functions". J. Comput. Phys. 92: 230. Bibcode:1991JCoPh..92..230C. doi:10.1016/0021-9991(91)90299-Z.
  6. ^ Blanes, S.; Moan, P. C. (May 2002). "Practical symplectic partitioned Runge–Kutta and Runge–Kutta–Nyström methods". Journal of Computational and Applied Mathematics. 142 (2): 313–330. Bibcode:2002JCoAM.142..313B. doi:10.1016/S0377-0427(01)00492-7.
  7. ^ Tao, Molei (2016). "Explicit symplectic approximation of nonseparable Hamiltonians: Algorithm and long time performance". Phys. Rev. E. 94: 043303. arXiv:1609.02212. Bibcode:2016PhRvE..94d3303T. doi:10.1103/PhysRevE.94.043303.
  8. ^ Qin, H.; Guan,X. (2008). "A Variational Symplectic Integrator for the Guiding Center Motion of Charged Particles for Long Time Simulations in General Magnetic Fields" (PDF). Physical Review Letters. 100: 035006. doi:10.1103/PhysRevLett.100.035006.
  9. ^ Qin, H.; Liu, J.; Xiao,J. (2016). "Canonical symplectic particle-in-cell method for long-term large-scale simulations of the Vlasov–Maxwell equations". Nuclear Fusion. 56: 014001. arXiv:1503.08334. doi:10.1088/0029-5515/56/1/014001.
  10. ^ Zhang, R.; Qin, H.; Tang, Y. (2016). "Explicit symplectic algorithms based on generating functions for charged particle dynamics". Physical Review E. 94: 013205. arXiv:1604.02787. doi:10.1103/PhysRevE.94.013205.
  11. ^ He, Y.; Qin, H.; Sun, Y. (2015). "Hamiltonian integration methods for Vlasov-Maxwell equations". Physics of Plasmas. 22: 124503. arXiv:1505.06076. doi:10.1063/1.4938034.
  12. ^ Xiao, J.; Qin, H.; Liu, J. (2015). "Explicit high-order non-canonical symplectic particle-in-cell algorithms for Vlasov-Maxwell systems". Physics of Plasmas. 22: 112504. arXiv:1510.06972. doi:10.1063/1.4935904.
  13. ^ Kraus, M; Kormann, K; Morrison, P.; Sonnendrucker, E (2017). "GEMPIC: geometric electromagnetic particle-in-cell methods". Journal of Plasma Physics. 83: 905830401. arXiv:1609.03053. doi:10.1017/S002237781700040X.
  14. ^ Xiao, J.; Qin, H.; Liu, J. (2018). "Structure-preserving geometric particle-in-cell methods for Vlasov-Maxwell systems". Plasma Science and Technology. 20: 110501. arXiv:1804.08823. doi:10.1088/2058-6272/aac3d1.
  • Leimkuhler, Ben; Reich, Sebastian (2005). Simulating Hamiltonian Dynamics. Cambridge University Press. ISBN 0-521-77290-7.
  • Hairer, Ernst; Lubich, Christian; Wanner, Gerhard (2006). Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations (2 ed.). Springer. ISBN 978-3-540-30663-4.