论文标题
FILON方法中的递归力矩计算，并应用于二维的高频波散射
Recursive moment computation in Filon methods and application to high-frequency wave scattering in two dimensions
论文作者
论文摘要
我们使用FILON方法研究了高度振荡积分的有效近似。实现这些方法的关键步骤是对符号正交矩的准确，快速计算。在这项工作中，我们证明了如何基于观察到的观察结果，即许多物理相关的核函数在线性差分算子的无空间空间中，其在fileon插值基础上的作用以频带（infinite）矩阵表示。我们进一步详细讨论了对两个特定兴趣类别的应用程序，具有代数奇点的积分以及涉及Hankel函数的固定点和积分。我们为这些类别的第一个类别的计算提供了严格的稳定性结果，并演示了相应的FILON方法如何以真正独立的频率成本产生准确的近似值。对于Hankel内核，我们得出了错误估计，该估计描述了该方法在频率和数值正交点的数量方面的收敛行为。最后，我们展示了如何应用具有递归力矩计算的FILON方法来计算有效的积分，这些方法是在屏幕上高频波散射的混合数值 - 反对拼接方法中产生的。
We study the efficient approximation of highly oscillatory integrals using Filon methods. A crucial step in the implementation of these methods is the accurate and fast computation of the Filon quadrature moments. In this work we demonstrate how recurrences can be constructed for a wide class of oscillatory kernel functions, based on the observation that many physically relevant kernel functions are in the null space of a linear differential operator whose action on the Filon interpolation basis is represented by a banded (infinite) matrix. We discuss in further detail the application to two classes of particular interest, integrals with algebraic singularities and stationary points and integrals involving a Hankel function. We provide rigorous stability results for the moment computation for the first of these classes and demonstrate how the corresponding Filon method results in an accurate approximation at truly frequency-independent cost. For the Hankel kernel, we derive error estimates which describe the convergence behaviour of the method in terms of frequency and number of Filon quadrature points. Finally, we show how Filon methods with recursive moment computation can be applied to compute efficiently integrals arising in hybrid numerical-asymptotic collocation methods for high-frequency wave scattering on a screen.
