Symplectic Integration Schemes Research Articles

An Explicit Symplectic Integration Scheme for Plasma Simulations

An explicit symplectic integration scheme which describes the self-consistent wave particle interaction is developed. The integrator does not split the hamiltonian trivially into a kinetic and potential part. The integrator yields accurate growth rates for the gentle-bump instability even when the timestep is of the order of the inverse plasma frequency. This represents up to a tenfold reduction in computation compared to conventional schemes. The integrator is generalizable to arbitrary order without increase in storage requirements, but tests show that when the accuracy requirements are of the order of a few percent, the second-order method is the most efficient.

Approximations to solitary waves on lattices. II. quasi-continuum methods for fast and slow waves

A number of highly accurate analytic approximations to solitary waves on lattices are detailed, and the results of numerical tests presented. Two types of approximation are considered. The first set of approximations are generalizations of the continuum approximation, based around various Pade expansions of the operator. These are convergent to the exact solution in the limit of wavespeed c to 1, or a dense lattice. The author also considers ways of constructing approximations to fast solitary waves from a continuum-type theory. The two types are closely related. In both cases significant improvements are made on existing approximations. A symplectic Hamiltonian integration scheme is used to perform computer simulations. A new method of measuring the accuracy of a predicted waveform is also used on new and old approximations alike, to compare the various methods considered.

The discrete variational approach to the Euler-Lagrange equation

We apply symplectic integration schemes to solve two-point boundary value problems for the Euler-Lagrange equation. These methods admit a natural discrete variational principle, and are, therefore, a new type of approximation. Numerical results are given for several examples.