In mathematics, a symplectic integrator (SI) is a numerical integration scheme for Hamiltonian systems. Symplectic integrators form the subclass of geometric integrators which, by definition, are canonical transformations. They are widely used in nonlinear dynamics, molecular dynamics, discrete element methods, accelerator physics, plasma physics, quantum physics, and celestial mechanics.
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.
Methods for constructing symplectic algorithms
Splitting methods for separable Hamiltonians
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
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
Note that both of these maps are practically computable.
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:
- 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 example
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 example
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 example
A fourth-order example
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. This fourth-order integrator was published in 1990 by Forest and Ruth and also independently discovered by two other groups around that same time.
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 further developed partitioned Runge–Kutta methods for the integration of systems with separable Hamiltonians with very small error constants.
Splitting methods for general nonseparable Hamiltonians
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. 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.
In Plasma Physics
In recent decades symplectic integrator in plasma physics has become an active research topic, 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. 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.
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. Splitting into 4 parts,
we find serendipitously that for each subsystem, e.g.,
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
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.