论文标题
傅里叶平滑的预校正的梯形规则，用于解决Lippmann-Schwinger积分方程的解决方案
Fourier smoothed pre-corrected trapezoidal rule for solution of Lippmann-Schwinger integral equation
论文作者
论文摘要
对于Lippmann-Schwinger方程的数值解，而预校正的梯形规则则以高阶收敛，以使平滑的紧凑型密度平滑，在整个界面材料属性中不连续性的情况下，它仅表现出线性收敛。 在这篇简短的文章中，我们提出了一个基于“傅里叶平滑的预校正的梯形规则”的Nyström求解器，该规则与二阶收敛，以解决此类散射问题，同时保持$ O（n \ log n）$的计算复杂性。此外，该方法不仅非常易于实现，还适用于几何复杂的不均匀性问题，包括那些具有角落和尖端的问题。我们提出了多种数值实验，包括与[J.计算。 Phys。，200（2）（2004），670--694] Bruno and Hyde，以及[J.傅立叶肛门。 Appl。，11（4）（2005），471-487] Andersson和Holst在速度和准确性方面举例说明了其性能。这种傅立叶平滑的数值集成方案也可以适应其他感兴趣的问题，其中需要计算具有不连续密度的卷积积分。
For the numerical solution of the Lippmann-Schwinger equation, while the pre-corrected trapezoidal rule converges with high-order for smooth compactly supported densities, it exhibits only the linear convergence in the case of discontinuity in material properties across the interface. In this short article, we propose a Nyström solver based on "Fourier smoothed pre-corrected trapezoidal rule" that converges with second order for such scattering problems while maintaining the computational complexity of $O(N \log N)$. Moreover, the method is not only very simple to implement, it is also applicable to problems with geometrically complex inhomogeneities including those with corners and cusps. We present a variety of numerical experiments including comparative studies with competing approaches reported in [J. Comput. Phys., 200(2) (2004), 670--694] by Bruno and Hyde, and in [J. Fourier Anal. Appl., 11(4) (2005), 471-487 ] by Andersson and Holst to exemplify its performance in terms of speed and accuracy. This Fourier smoothed numerical integration scheme can also be adapted to other problems of interest where the convolution integral with discontinuous density is required to be computed.
