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：我们提出了一种基于Neumann级数和Filon求积的三阶数值积分器，主要针对高度振荡偏微分方程设计。该方法可应用于呈现小幅度或中等幅度振荡的方程；然而反直觉的是，大幅振荡反而能提高该方案的精度。通过所提出的方法，可以轻松提高方法的收敛阶数。我们还对该方法进行了误差分析。我们考虑了包含一阶和二阶时间导数的线性发展方程，这些方程涉及椭圆型微分算子，例如热传导方程或波动方程。数值实验考虑了空间维数大于1的情况，并证实了理论分析结果。