Symplectic integrators for the matrix Hill's equation and its applications to engineering models

TL;DR

This paper introduces high-order symplectic exponential integrators specifically designed for the matrix Hill's equation, preserving its Hamiltonian structure and efficiently handling oscillatory solutions in physical models.

Contribution

The paper develops new sixth- and eighth-order symplectic exponential integrators tailored for the matrix Hill's equation, enhancing accuracy and efficiency in numerical simulations.

Findings

The new integrators accurately solve oscillatory problems with low computational cost.

Numerical examples demonstrate superior performance over existing methods.

The methods effectively preserve the symplectic structure of the solution.

Abstract

We consider the numerical integration of the matrix Hill's equation. Parametric resonances can appear and this property is of great interest in many different physical applications. Usually, the Hill's equations originate from a Hamiltonian function and the fundamental matrix solution is a symplectic matrix. This is a very important property to be preserved by the numerical integrators. In this work we present new sixth-and eighth-order symplectic exponential integrators that are tailored to the Hill's equation. The methods are based on an efficient symplectic approximation to the exponential of high dimensional coupled autonomous harmonic oscillators and yield accurate results for oscillatory problems at a low computational cost. Several numerical examples illustrate the performance of the new methods.