Third order, uniform in low to high oscillatory coefficients, exponential integrators for Klein-Gordon equations

TL;DR

This paper introduces a third-order exponential integrator for Klein-Gordon equations with space- and time-dependent mass, effectively handling high oscillations with large time steps, and demonstrates its convergence and robustness through theoretical analysis and numerical tests.

Contribution

It develops a novel third-order exponential integrator that embeds oscillations into the discretization, enabling large time steps even with high oscillatory frequencies.

Findings

The integrator achieves high accuracy without error growth at increasing oscillation frequencies.

Numerical simulations confirm the method's robustness across various oscillatory regimes.

Theoretical convergence proofs support the effectiveness of the proposed approach.

Abstract

Allowing for space- and time-dependence of mass in Klein--Gordon equations resolves the problem of negative probability density and of violation of Lorenz covariance of interaction in quantum mechanics. Moreover it extends their applicability to the domain of quantum cosmology, where the variation in mass may be accompanied by high oscillations. In this paper we propose a third-order exponential integrator, where the main idea lies in embedding the oscillations triggered by the possibly highly oscillatory component intrinsically into the numerical discretisation. While typically high oscillation requires appropriately small time steps, an application of Filon methods allows implementation with large time steps even in the presence of very high oscillation. This greatly improves the efficiency of the time-stepping algorithm. Proof of the convergence and its rate are nontrivial and require alternative representation of the equation under consideration. We derive careful bounds on the growth of global error in time discretisation and prove that, contrary to standard intuition, the error of time integration does not grow once the frequency of oscillations increases. Several numerical simulations are presented to confirm the theoretical investigations and the robustness of the method in all oscillatory regimes.