NSI-IBP: A General Numerical Singular Integral Method via Integration by Parts

TL;DR

The NSI-IBP framework reframes singular and nearly singular integrals via Integration by Parts by introducing a surrogate $q(x)$ with known inverse integral $h(x)$ to convert a challenging integral $I_0= int_a^b\frac{p(x)}{q(x)}dx$ into a non-singular remainder plus boundary terms. When the exact $q(x)$ is unknown, a surrogate $ ilde{q}(x)$ is used to form a computable IBP expression, with error controlled by the mismatch between $ ilde{q}$ and $q$; the method also recovers conventional IBP in the special case $ ilde{q}=q$. The authors present a unifying general IBP formula that encompasses unknown and known $q(x)$ cases, derive error bounds, and provide a practical numerical recipe. Through extensive numerical experiments on power-law, logarithmic, and hybrid singularities, and on nearly singular Green’s-function integrals in electrostatics and CEM, NSI-IBP achieves relative accuracies down to $10^{-15}$ in favorable settings and remains robust when the precise singular form is not known. While demonstrated in one dimension for clarity, the approach extends to higher dimensions by decomposing complex integrals into multiple 1D problems, offering a promising tool for efficient, accurate computation of challenging singular and near-singular integrals in physics and engineering.

Abstract

A general framework of Numerical Singular Integrals (NSI) method based on the Integration By Parts (IBP) has been developed for integrals involving singular and nearly singular integrands, or NSI-IBP. Through a general integration by parts formula and by choosing some analytically integrable function to approximate the original integrand, various well-known integration by parts methods can be derived. Rigorous mathematical derivations have been performed to transform the original singular or nearly singular integrals into non-singular integrals that can be computed efficiently, along with the boundary values added. What's more important, the NSI-IBP method works well even when the exact form of the singular integrand is not known. Criteria on how to choose the appropriate function with a known analytical integral that closely approximates the original integrand have been outlined and explained. Numerical recipe has been presented to apply the proposed NSI-IBP. Numerical experiments have been carried out on various singular integrals such as the power-law decaying integrand, the logarithmic function, and their hybrid products. It can be shown that various relative accuracy up to $10^{-15}$ can be achieved, even the exact singular function is not known. Finally, the nearly singular integrals involving the scalar Green's function have been evaluated for both electrostatics applications and Computational Electromagnetics (CEM) applications.

Figures (9)

• Figure 1: The convolution integral of scalar Green's function shown together with the RWG surface current on a pair of triangles. The red points denote the observation locations. Without loss of generality, the right triangles of unit length is used for the numerical experiment because any quadrilateral can be mapped to the unit square using bilinear transformation.
• Figure 2: Accuracy surface plot for the power-law singular function of $\gamma = 1/2$, with respect to the deviation of $\Delta \gamma \in [10^{-12}, 1] \gamma$ for different offsets $o \in [10^{-25}, 10^4]$.
• Figure 3: Accuracy lines plot for the power-law singular function with $\gamma = 1/2$, for different offset deviation $o \in [10^{-25}, 10^4]$, when different deviations of $\Delta \gamma \in [10^{-12}, 1] \gamma$ are used.
• Figure 4: Accuracy surface plot for the hybrid singular function of $\gamma = 1/2$, with respect to the deviations of $\Delta$ for different offsets $o$.
• Figure 5: Accuracy lines plot for the hybrid singular function with $\gamma = [0, 9]$, for different offsets $o \in [10^{-25}, 10^4]$, when different deviations $\Delta \gamma \in [10^{-12}, 1] \gamma$ are used.