SinCoTrap：任意维度下周期奇异积分的高阶局部修正梯形法则
SinCoTrap: A High-Order Locally Corrected Trapezoidal Rule for Periodic Singular Integrals in Arbitrary Dimensions
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中文总结 AI 辅助
研究任意维度下周期奇异积分，提出SinCoTrap方法，通过保留均匀张量网格并修改奇点附近权重模板，利用解析延拓导出公式，使该方法在应用中高效且对高阶精度鲁棒，误差率达\(O(h^{2p + 2 + d - s})\)。
中文摘要 AI 辅助
我们提出了SinCoTrap（奇异校正梯形法则），这是一种用于任意维度\(d\)且核为\(|\boldsymbol{x}|^{-s}\)（\(0<s<d\)）的周期奇异积分的高阶局部修正梯形方法。该方案保留均匀张量网格，仅修改奇点附近固定的小权重模板。对于校正阶数\(p\)，所得求积法的误差率为\(O(h^{2p + 2 + d - s})\)。我们通过黎曼zeta函数的特殊推广的解析延拓导出明确的、与网格无关的极限校正权重，得到可快速计算的公式，可针对每个\((d,s,p)\)预先制表。这使得SinCoTrap在应用中高效且在广泛的周期奇异积分中对高阶精度具有鲁棒性。
英文摘要
We present SinCoTrap (Singularity-Corrected Trapezoidal Rule), a high-order locally corrected trapezoidal method for periodic singular integrals in arbitrary dimension $d$ with kernel $|\boldsymbol{x}|^{-s}$, $0<s<d$. The scheme preserves the uniform tensor grid and modifies only a fixed, small stencil of weights near the singularity. For a correction order $p$, the resulting quadrature attains the error rate $O(h^{2p+2+d-s})$. We derive explicit, mesh-independent limiting correction weights via analytic continuation of a special generalization of the Riemann zeta function, yielding rapidly computable formulas that can be pretabulated for each $(d,s,p)$. This makes SinCoTrap both efficient in application and robust for high-order accuracy across a broad class of periodic singular integrals.
