We present a fast and accurate numerical scheme for approximating weakly singular oscillatory Volterra integral equations (VIEs) of the second kind. The main idea is the combination of the black-box fast multipole method (FMM) and collocation methods that leads to a significant speed-up in CPU time for a given tolerance. The weakly singular oscillatory integrals, which arise from the evaluation of integral equations, are calculated using Clenshaw-Curtis Filon methods. Because the kernel function is approximated by a Chebyshev interpolation scheme, it is very useful for the kernel which is known analytically but is quite complicated. Compared with the method and the error estimates reported previously, the new scheme is suitable for more general kernel function and provide sharper estimates. The efficiency and accuracy of the new method are verified by numerical examples.

我们提出了一种快速准确的数值方案，用于逼近第二类弱奇异振荡沃尔泰拉积分方程 (VIE)。主要思想是将黑盒快速多极子方法 (FMM) 和配置方法相结合，从而在给定容差下显着加快 CPU 时间。使用 Clenshaw-Curtis Filon 方法计算由积分方程的评估产生的弱奇异振荡积分。因为核函数是通过切比雪夫插值方案逼近的，所以对于解析已知但相当复杂的核非常有用。与之前报道的方法和误差估计相比，新方案适用于更一般的核函数并提供更清晰的估计。