Effective highly accurate time integrators for linear Klein-Gordon equations across the scales

TL;DR

This paper introduces novel highly accurate time integrators for linear Klein-Gordon equations that efficiently handle a wide range of oscillatory regimes without error growth, demonstrated through numerical experiments.

Contribution

The paper presents a new class of splitting-based time integrators specifically designed for highly oscillatory Klein-Gordon equations, maintaining accuracy across all frequency regimes.

Findings

Methods achieve uniform accuracy regardless of oscillation frequency

Numerical experiments confirm theoretical error bounds

New schemes outperform existing methods in efficiency and accuracy

Abstract

We propose an efficient approach for time integration of Klein-Gordon equations with highly oscillatory in time input terms. The new methods are highly accurate in the entire range, from slowly varying up to highly oscillatory regimes. Our approach is based on splitting methods tailored to the structure of the input term which allows us to resolve the oscillations in the system uniformly in all frequencies, while the error constant does not grow as the oscillations increase. Numerical experiments highlight our theoretical findings and demonstrate the efficiency of the new schemes.