Effective highly accurate integrators for linear Klein-Gordon equations from low to high frequency regimes

We introduce an efficient class of numerical schemes for the Klein--Gordon equation which are highly accurate from slowly varying up to highly oscillatory regimes. Their construction is based on Magnus expansions tailored to the structure of the input term which allows us to resolve the oscillations in the system up to second order convergence in time uniformly in all frequencies $\omega_n$. Depending on the nature of the oscillatory term, the proposed methods even show superior convergence, reaching up to fourth-order convergence, while maintaining high efficiency and small error constants. Numerical experiments highlight our theoretical findings and underline the efficiency of the new schemes.