AI 中文总结
本文提出并分析了一种适用于二维周期光栅衍射的频率稳定Nyström方法，通过修正准周期格林函数克服Wood异常频率处的奇异性，证明了离散系统的频率一致可解性与误差估计，数值例子验证了其稳定性。
AI 中文摘要
我们提出并分析了一种用于二维周期光栅标量波散射的频率稳定边界积分方程方法。标准准周期格林函数在Wood异常频率处作为频率函数是奇异的，这阻碍了对传统积分方程离散化进行频率一致的误差分析。我们的方法基于修正的准周期格林函数，该函数通过减去奇异的瑞利模式并引入有限维修正来恢复辐射条件得到。所得的组合场边界积分方程在基础衍射问题唯一可解时也被证明是唯一可解的，且对应的散射场在包括Wood异常频率在内的整个频率范围内连续依赖于频率。随后，我们利用格林函数的Ewald表示和适配于核函数对数奇点的求积规则构建了完全离散的Nyström格式。对于可能包含Wood异常频率的紧凑频率区间，我们证明了当离散化和截断参数足够大时离散系统的唯一可解性，并建立了频率一致的误差估计和条件数界。数值例子验证了该格式的收敛性及其在Wood异常频率处的稳定性。
英文摘要
We propose and analyze a frequency-stable boundary integral equation method for scalar wave scattering by periodic diffraction gratings in two dimensions. The standard quasi-periodic Green function is singular as a function of the frequency at Wood anomaly frequencies, which prevents a direct frequency-uniform error analysis of conventional integral equation discretizations. Our method is based on a modified quasi-periodic Green function obtained by subtracting the singular Rayleigh modes and introducing a finite-dimensional correction that restores the radiation condition. The resulting combined-field boundary integral equation is shown to be uniquely solvable whenever the underlying diffraction problem is uniquely solvable, and the corresponding scattered field depends continuously on the frequency, including at Wood anomalies. We then construct a fully discrete Nyström scheme using an Ewald representation of the Green function and a quadrature rule adapted to the logarithmic singularity of the kernel. For compact frequency intervals that may contain Wood anomaly frequencies, we prove unique solvability of the discrete system for sufficiently large discretization and truncation parameters and establish a frequency-uniform error estimate and bound on the condition number. Numerical examples illustrate the convergence of the scheme and its stability across Wood anomaly frequencies.
Comments27 pages, 1 figure
