Symplectic integrator

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In mathematics, a symplectic integrator (SI) is a numerical integration scheme for a specific group of differential equations related to classical mechanics. Symplectic integrators are a class of geometric integrators. They are widely used in molecular dynamics, discrete element methods and celestial mechanics.

[edit] Introduction

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

\dot p = -\frac{\partial H}{\partial q} \quad\mbox{and}\quad \dot q = \frac{\partial H}{\partial p},

where q denotes the position coordinates, p the momentum coordinates, and H is the Hamiltonian (see Hamiltonian mechanics for more background).

The time evolution of Hamilton's equations is a symplectomorphism, meaning that it conserves the symplectic two-form dp \wedge dq. 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.

[edit] 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

H(p,q) = T(p) + V(q). \qquad\qquad (1)

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

Then the equations of motion of a Hamiltonian system can be expressed as

\dot{z}=\{z,H(z)\} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)

where \{\cdot, \cdot\} is a Poisson bracket. By using the notation D_H = \{\cdot, H\}, this can be re-expressed as

\dot{z}=D_H z.

The formal solution of this set of equations is given as

z(\tau)=\exp(\tau D_H)z(0). \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)

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

z(\tau) = \exp[\tau (D_T + D_V)]z(0). \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (4)

The SI scheme approximates the time-evolution operator exp[τ(DT + DV)] in the formal solution (4) by a product of operators as

\exp[\tau (D_T + D_V)] = \Pi_{i=1}^k \exp(c_i \tau D_T)\exp(d_i \tau D_V) + O(\tau^{n+1}), \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (5)

where ci and di are real numbers, and n is an integer, which is called the order of the integrator. Note that each of the operators exp(ciτDT) and exp(diτDV) provides a symplectic map, so their product appearing in the right hand side of (5) also constitutes a symplectic map. In concrete terms, exp(ciτDT) gives the mapping

\begin{pmatrix} q\\ p \end{pmatrix} \mapsto \begin{pmatrix} q'\\ p' \end{pmatrix} = \begin{pmatrix} q + \tau c_i \frac{\partial T}{\partial p}(p)\\ p \end{pmatrix}

and exp(diτDV) gives

\begin{pmatrix} q\\ p \end{pmatrix} \mapsto \begin{pmatrix} q'\\ p' \end{pmatrix} = \begin{pmatrix} q \\ p - \tau d_i \frac{\partial V}{\partial q}(q)\\ \end{pmatrix}.

Note that both of these maps are practically computable.

The coefficients for the first-order integrator (k = 1) are simply

c_1 = d_1 = 1. \ \

The coefficients for a second-order integrator (k = 2) are given by

c_1 = c_2 = 1/2, \ \
d_1 = 1,\ \ d_2 = 0

A fourth order integrator (k = 4) was independently discovered by three groups [1] [2] [3]

c_1 = c_4 = \frac{1}{2(2-2^{1/3})},\ \ c_2=c_3=\frac{1-2^{1/3}}{2(2-2^{1/3})},
d_1 = d_3 = \frac{1}{2-2^{1/3}},\ \ d_2 = -\frac{2^{1/3}}{2-2^{1/3}},\ \ d_4 = 0

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.

[edit] References

  1. ^ Forest, E., Ruth, R.D. (1990). "Fourth-order symplectic integration". Physica D 43: 105.
  2. ^ Yoshida, H. (1990). "Construction of higher order symplectic integrators". Phys. Lett. A 150: 262.
  3. ^ Candy, J., Rozmus, W. (1991). "A Symplectic Integration Algorithm for Separable Hamiltonian Functions". J. Comput. Phys. 92: 230.
  • Leimkuhler, Ben, Sebastian Reich (2005). Simulating Hamiltonian Dynamics. Cambridge University Press. ISBN 0-521-77290-7.
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