We propose a third-order numerical integrator based on the Neumann series and the Filon quadrature, designed mainly for highly oscillatory partial differential equations. The method can be applied to equations that exhibit small or moderate oscillations; however, counter-intuitively, large oscillations increase the accuracy of the scheme. With the proposed approach, the convergence order of the method can be easily improved. Error analysis of the method is also performed. We consider linear evolution equations involving first- and second-time derivatives that feature elliptic differential operators, such as the heat equation or the wave equation. Numerical experiments consider the case in which the space dimension is greater than one and confirm the theoretical study.

翻译：本文提出一种基于诺伊曼级数和菲隆求积法的三阶数值积分器，主要面向高振荡偏微分方程设计。该方法适用于呈现弱振荡或中等振荡的方程；然而反直觉的是，强振荡反而能提升该格式的精度。通过所提出的方法，可轻松改进该数值方法的收敛阶数。本文同时进行了该方法的误差分析。我们考察了涉及一阶与二阶时间导数、且具有椭圆型微分算子的线性演化方程，例如热传导方程或波动方程。数值实验考虑了空间维度大于一的情形，结果验证了理论研究的正确性。