Wikipedia / Symplectic integrator
Wikipedia / Symplectic integrator
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.
Introduction
Symplectic integrators are designed for the numerical solution of Hamilton's equations 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 2-form . A numerical scheme is a symplectic integrator if it also conserves this 2-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, such as the primitive Euler scheme and the classical Runge–Kutta scheme, are not symplectic integrators.
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
This happens frequently in Hamiltonian mechanics, with being the kinetic energy and the potential energy.
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
is a Poisson bracket, which returns a Poisson bracket of the operand with the Hamiltonian, the expression of the Hamilton's equation can be further simplified to
is given as a matrix exponential:
in the matrix exponential.
When the Hamiltonian has the form of equation (), the solution () is equivalent to
in the formal solution () by a product of operators as
provides a symplectic map, so their product appearing in the right-hand side of () also constitutes a symplectic map.
, we can conclude that
By using a Taylor series can be expressed as
is an arbitrary real number. Combining () and (, we get
gives the mapping
gives
Note that both of these maps are practically computable.
Algorithm
:
is the scalar quantity of mass.
Equivalently, after converting into Lagrangian coordinates, each substep becomes
.
Several symplectic integrators are given below.
A first-order example
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 and coefficients
Note that the coefficients
also form a second-order symplectic integrator.
in the second set of coefficients, in both algorithms above one update can be ignored. There are thus only 3 steps to the algorithm, and step 1 and 3 are exactly the same, leading to the algorithm being symmetric in time, so the positive time version can also be used for negative time.
A third-order example
) was discovered by Ronald Ruth in 1983.
A fourth-order example
) 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. Numerov%27s method, developed by the Russian astronomer Boris Vasil'evich Numerov in 1924, is also a fourth-order symplectic integrator.
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. in the sense that
corresponds to a linear transformation. To symplectically simulate the system, one simply composes these solution maps.
Applications
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 the physics problem under investigation. One such example is the charged particle dynamics in an electromagnetic field is actually explicit. This is what is used in the canonical symplectic particle-in-cell (PIC) algorithm. are product-separable, 2nd and 3rd order explicit symplectic algorithms can be constructed using generating functions, and arbitrarily high-order explicit symplectic integrators for time-dependent electromagnetic fields can also be constructed using Runge-Kutta techniques.
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. which is a customarily modified Strang splitting The He splitting method is one of key techniques used in the structure-preserving geometric particle-in-cell (PIC) algorithms.
See also
References