Symplectic integrators are designed for the numerical solution of Hamilton's equations, which read where
The time evolution of Hamilton's equations is a symplectomorphism, meaning that it conserves the symplectic 2-form
Symplectic integrators possess, as a conserved quantity, a Hamiltonian which is slightly perturbed from the original one.
[1] 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.
A widely used class of symplectic integrators is formed by the splitting methods.
, 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
When the Hamiltonian has the form of equation (1), the solution (3) is equivalent to The SI scheme approximates the time-evolution operator
Note that due to the definitions adopted above (in the operator version of the explanation), the index
in decreasing order): Iteratively: The symplectic Euler method is the first-order integrator with
) was also discovered by Ruth in 1983 and distributed privately to the particle-accelerator community at that time.
[3] This fourth-order integrator was published in 1990 by Forest and Ruth and also independently discovered by two other groups around that same time.
Later on, Blanes and Moan[7] further developed partitioned Runge–Kutta methods for the integration of systems with separable Hamiltonians with very small error constants.
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 new Hamiltonian is advantageous for explicit symplectic integration, because it can be split into the sum of three sub-Hamiltonians,
solutions correspond to shifts of mismatched position and momentum, and
To symplectically simulate the system, one simply composes these solution maps.
In recent decades symplectic integrator in plasma physics has become an active research topic,[9] 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.
-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.
-dependence are entangled in this Hamiltonian, and try to design a symplectic algorithm just for this or this type of problem.
-dependence is quadratic, therefore the first order symplectic Euler method implicit in
[10] To build high order explicit methods, we further note that the
are product-separable, 2nd and 3rd order explicit symplectic algorithms can be constructed using generating functions,[11] and arbitrarily high-order explicit symplectic integrators for time-dependent electromagnetic fields can also be constructed using Runge-Kutta techniques.
[12] A more elegant and versatile alternative is to look at the following non-canonical symplectic structure of the problem,
However, for this specific problem, a family of high-order explicit non-canonical symplectic integrators can be constructed using the He splitting method.
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.
The He splitting method is one of key techniques used in the structure-preserving geometric particle-in-cell (PIC) algorithms.