SinCoTrap: A High-Order Locally Corrected Trapezoidal Rule for Periodic Singular Integrals in Arbitrary Dimensions

Abstract

We present SinCoTrap (Singularity-Corrected Trapezoidal Rule), a high-order locally corrected trapezoidal method for periodic singular integrals in arbitrary dimension $d$ with kernel $|\mathbf{x}{|}^{-s}$, $0<s<d$. 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.