自适应Filon型方法在高振荡积分中的数值研究

Numerical Study of Adaptive Filon Method in High Oscillation Integral

摘要: 高振荡积分广泛存在于波动传播、信号分析、量子物理和工程计算等领域。针对传统数值积分方法在高频振荡条件下采样需求大、计算精度不稳定的问题，本文研究一类Fourier型高振荡积分的数值计算方法。在Filon型求积框架下，对振幅函数进行局部多项式逼近，并对振荡因子作解析积分处理，以减弱高频振荡对数值计算的影响。在此基础上，引入基于振幅函数变化指标的自适应分段策略，使节点在变化剧烈区域加密、在平缓区域保持稀疏，从而提高节点利用效率。数值实验选取指数型、有理型和局部尖峰型振幅函数，将复合梯形公式、复合Simpson公式、复合Gauss-Legendre求积、普通等距Filon方法与自适应分段Filon方法进行对比。结果表明，在较高振荡频率下，Filon型方法较传统直接求积方法更稳定；自适应分段策略能够有效降低局部插值误差，并改善普通等距Filon方法的计算精度。

Abstract: Highly oscillatory integrals arise widely in wave propagation, signal analysis, quantum physics, and engineering computation. This paper studies numerical methods for a class of Fourier-type oscillatory integrals. Using the Filon quadrature framework, the amplitude function is locally approximated by polynomials, while the oscillatory factor is integrated analytically to reduce the effect of high-frequency oscillations. An adaptive partition strategy is further introduced according to the variation of the amplitude function, refining nodes in rapidly changing regions and keeping them sparse in smooth regions. Numerical experiments on exponential, rational, and locally peaked amplitude functions compare the proposed adaptive Filon method with the composite trapezoidal rule, composite Simpson rule, composite Gauss-Legendre quadrature, and uniform Filon methods. The results show that Filon-type methods are more stable than direct quadrature methods at high frequencies, and the adaptive strategy effectively reduces local interpolation errors and improves the accuracy of the standard uniform Filon method.