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.

翻译：在Klein-Gordon方程中允许质量随空间和时间变化，解决了量子力学中负概率密度和Lorenz协变性违背的问题。同时，这将其应用范围扩展到量子宇宙学领域，其中质量的变化可能伴随高频振荡。本文提出了一种三阶指数积分器，其核心思想在于将可能由高度振荡分量触发的振荡内嵌于数值离散化过程中。虽然高振荡通常需要适当小的步长，但Filon方法的应用使得即使在存在极高振荡的情况下也能采用大步长实现。这极大提高了时间步进算法的效率。收敛性及其收敛速度的证明具有非平凡性，需对所述方程进行替代表示。我们推导了时间离散化中全局误差增长的严格界限，并证明与常规直觉相反，时间积分的误差不会随振荡频率增加而增长。通过多项数值模拟验证了理论分析结果及该方法在所有振荡机制下的鲁棒性。