Decomposition and conformal mapping techniques for the quadrature of nearly singular integrals

TL;DR

This paper explores advanced decomposition and conformal mapping methods to efficiently compute nearly singular integrals, especially when singularity locations are known, with applications to viscous flow surface integrals.

Contribution

It generalizes existing techniques for improving quadrature of nearly singular integrals using decomposition and conformal mapping, tailored for known singularity positions.

Findings

Enhanced quadrature accuracy for nearly singular integrals

Reduced computational time compared to standard methods

Successful application to viscous flow surface integrals

Abstract

Gauss-Legendre quadrature and the trapezoidal rule are powerful tools for numerical integration of analytic functions. For nearly singular problems, however, these standard methods become unacceptably slow. We discuss and generalize some existing methods for improving on these schemes when the location of the nearby singularity is known. We conclude with an application to some nearly singular surface integrals of viscous flow.