A general correction for numerical integration rules over piece-wise continuous functions

TL;DR

This paper introduces a correction method for classical quadrature rules that improves their accuracy when integrating functions with known discontinuities by accounting for jumps in the function and its derivatives.

Contribution

The paper proposes a novel correction approach that enhances classical numerical integration rules to handle piece-wise continuous functions with discontinuities effectively.

Findings

Correction terms improve integration accuracy near discontinuities

Method maintains accuracy of classical rules for piece-wise functions

Numerical experiments validate theoretical improvements

Abstract

This article presents a novel approach to enhance the accuracy of classical quadrature rules by incorporating correction terms. The proposed method is particularly effective when the position of an isolated discontinuity in the function and the jump in the function and its derivatives at that position are known. Traditional numerical integration rules are exact for polynomials of certain degree. However, they may not provide accurate results for piece-wise polynomials or functions with discontinuities without modifying the location and number of data points in the formula. Our proposed correction terms address this limitation, enabling the integration rule to conserve its accuracy even in the presence of a jump discontinuity. The numerical experiments that we present support the theoretical results obtained.