# Symplectic integrators for second-order linear non-autonomous equations

## TL;DR

This paper introduces two new symplectic integrators derived from the Magnus expansion, tailored for second-order linear non-autonomous equations, with one family optimized for low to moderate dimensions and the other for large-scale systems.

## Contribution

The paper presents two novel symplectic methods based on the Magnus expansion, designed specifically for different problem sizes of second-order linear non-autonomous equations.

## Key findings

- Effective for low to moderate dimensions
- Suitable for large systems like discretized wave equations
- Numerical experiments demonstrate scheme advantages

## 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.