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.

我们提出了一种基于诺依曼级数和菲隆求积的三阶数值积分器，主要针对高振荡偏微分方程而设计。该方法可应用于表现出小或中等振荡的方程；然而，与直觉相反，大的振荡提高了该方案的准确性。通过所提出的方法，可以轻松地提高该方法的收敛阶数。还对该方法进行了误差分析。我们考虑涉及具有椭圆微分算子的一阶和二阶导数的线性演化方程，例如热方程或波动方程。数值实验考虑了空间维数大于1的情况，并证实了理论研究。