High order integrators obtained by linear combinations of symmetric-conjugate compositions

TL;DR

This paper introduces a new family of high-order numerical integrators using complex coefficients, constructed as linear combinations of symmetric-conjugate compositions, which preserve important geometric properties and are suitable for parallel computation.

Contribution

The paper develops a novel class of high-order integrators based on linear combinations of symmetric-conjugate compositions, extending the order and preserving symplecticity for Hamiltonian systems.

Findings

Methods up to order 10 were constructed and tested.

The new integrators outperform standard procedures in efficiency.

They preserve time-symmetry and symplecticity up to high order.

Abstract

A new family of methods involving complex coefficients for the numerical integration of differential equations is presented and analyzed. They are constructed as linear combinations of symmetric-conjugate compositions obtained from a basic time-symmetric integrator of order 2n (n $\ge$ 1). The new integrators are of order 2(n + k), k = 1, 2, ..., and preserve time-symmetry up to order 4n + 3 when applied to differential equations with real vector fields. If in addition the system is Hamiltonian and the basic scheme is symplectic, then they also preserve symplecticity up to order 4n + 3. We show that these integrators are well suited for a parallel implementation, thus improving their efficiency. Methods up to order 10 based on a 4th-order integrator are built and tested in comparison with other standard procedures to increase the order of a basic scheme.