论文标题
三阶，低至高振荡系数的均匀，klein-gordon方程的指数积分器
Third order, uniform in low to high oscillatory coefficients, exponential integrators for Klein-Gordon equations
论文作者
论文摘要
允许klein方程中质量的空间和时间依赖性解决了负概率密度和违反量子力学相互作用的洛伦兹协方差的问题。此外，它将其适用性扩展到量子宇宙学领域，在量子宇宙学的领域，质量变化可能伴随着高振荡。在本文中，我们提出了一个三阶指数积分器，其中主要思想在于嵌入可能由可能的高度振荡组件触发的振荡本质上触发的振荡中。虽然通常高振荡需要适当的时间步骤，但即使在存在很高振荡的情况下，FILON方法的应用也允许实现很大的时间步骤。这大大提高了时间步进算法的效率。 融合及其速率的证明是不平凡的，需要考虑所考虑的方程式的替代表示。我们在时间离散化中全局误差的增长谨慎界定，并证明与标准直觉相反，一旦振荡频率增加，时间整合的误差就不会增长。提出了几个数值模拟，以确认所有振荡制度中该方法的理论研究和鲁棒性。
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.
