Symplectic integrators for second-order linear non-autonomous equations

TL;DR

This paper develops two Magnus-based families of symplectic integrators for the second-order time-dependent linear system $x''(t) + M(t)x(t)=0$, addressing both moderate- and high-dimensional problems. Magnus-decomposition methods yield efficient matrix–matrix exponentials for small to moderate dimensions, while Magnus--splitting methods use matrix–vector products optimized for large-scale discretizations, such as linear time-dependent wave equations. A high-order 11-stage 6th-order method, $oldsymbol{
}_{11}^{[6]}$, is constructed with Gauss–Legendre quadrature and carefully chosen coefficients to ensure time symmetry and accuracy. Numerical experiments on the Mathieu equation, matrix Hill equation, and a trapped wave equation demonstrate improved accuracy-per-cost and robust long-time stability compared with standard Runge–Kutta and RKNyström schemes, especially in oscillatory and high-dimensional settings.

Abstract

Two families of symplectic methods specially designed for second-order time-dependent linear systems are presented. Both are obtained from the Magnus expansion of the corresponding first-order equation, but otherwise they differ in significant aspects. The first family is addressed to problems with low to moderate dimension, whereas the second is more appropriate when the dimension is large, in particular when the system corresponds to a linear wave equation previously discretised in space. Several numerical experiments illustrate the main features of the new schemes.

Figures (6)

• Figure 1: Error growth depending on $\omega$ in the Mathieu \ref{['ex:mathieu']} on a logarithmic scale.
• Figure 2: The performance of the 4th-order methods for the Mathieu \ref{['ex:mathieu']} on a $\log$--$\log$ scale; $cost = \mathcal{C}\times steps$.
• Figure 3: The performance of the 6th-order methods for the Mathieu \ref{['ex:mathieu']} on a $\log$--$\log$ scale; $cost = \mathcal{C}\times steps$.
• Figure 4: The performance of the 4th-order methods for the Hill \ref{['ex:hill']} on a $\log$--$\log$ scale; $cost = \mathcal{C}\times steps$.
• Figure 5: The performance of the 6th-order methods for the Mathieu \ref{['ex:hill']} on a $\log$--$\log$ scale; $cost = \mathcal{C}\times steps$.