Convergence of a class of high order corrected trapezoidal rules

TL;DR

This paper develops a convergence theory for corrected quadrature rules on uniform grids that accurately integrate functions with point singularities by using correction weights and series expansions, achieving high order accuracy.

Contribution

It introduces a novel framework for corrected trapezoidal rules that handle singularities with high order convergence, extending previous methods.

Findings

Error estimates for punctured trapezoidal rule derived

Corrected trapezoidal rules achieve high order accuracy near singularities

Series expansions enable effective correction for a broader class of functions

Abstract

We present convergence theory for corrected quadrature rules on uniform Cartesian grids for functions with a point singularity. We begin by deriving an error estimate for the punctured trapezoidal rule, and then derive error expansions. We define the corrected trapezoidal rules, based on the punctured trapezoidal rule, where the weights for the nodes close to the singularity are judiciously corrected based on these expansions. Then we define the composite corrected trapezoidal rules for a larger family of functions using series expansions around the point singularity and applying corrected trapezoidal rules appropriately. We prove that we can achieve high order accuracy by using a sufficient number of correction nodes around the point singularity and of expansion terms.