一些高振荡积分、高振荡积分方程的高性能计算

Efficient methods for some highly oscillatory integrals and integral equations

电磁、声波散射问题的研究涉及一类数学物理问题,此类问题具有深刻的理论价值和重要的应用背景,亟待解决.高振荡微分、积分方程是刻画这些问题的重要的数学模型,其数值计算存在许多挑战性研究课题.本文从积分方程解法角度出发,综述了求解这类高振荡问题的一些最新进展,特别是针对广义Fourier变换、Bessel变换的高效算法、高振荡核Volterra积分方程的数值解法作了详细介绍.这些数值方法共有特点是振荡频率越高算法精度愈高,且可望为电磁计算的研究提供一些新的高效算法.

A number of mathematical and physical problems occur in the areas of electromagnetics and acoustic scattering simulations, which are of important theoretical values and have wide applications. High oscillation plays a crucial role and represents formidable mathematical and computational challenge. Highly oscillatory differential equations and integral equations axe two fundamental models for these problems, whose computations are difficult and of many challenging problems. From the view of reformulation by means of integral equations, this paper gives a survey on the new developments on highly oscillatory problems, particularly, the details on generalized Fourier transforms, Bessel transforms and Volterra integral equations with highly oscillatory kernels. These methods share that the higher the frequency the more accurate of the numerical solution, which provides a new way of solving these kinds of equations.