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

TL;DR

The paper tackles the loss of quadrature accuracy for piecewise-smooth functions with isolated discontinuities by introducing explicit polynomial correction terms that leverage the known discontinuity location $x^*$ and jumps $[f^{(k)}]$. The correction is formulated for general quadrature and specifically tailored to Gaussian quadrature, with an exactness guarantee up to degree $2l+1$ for a single discontinuity. Numerical experiments across Newton-Cotes and Gauss-Legendre rules demonstrate that the corrected schemes achieve the intended higher-order convergence and, in Gauss-Legendre cases, can reproduce results to machine precision without increasing the number of nodes. The approach provides a general, efficient, and practical solution for integrating piecewise smooth functions in scenarios where discontinuity information is available.

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.

Figures (6)

• Figure 1: In this figure we can see an example of an isolated discontinuity in the function and its derivatives placed in the interval $[a,b]$ at a position $x^*$.
• Figure 2: Functions presented in (\ref{['function']}), (\ref{['function_corner']}), and (\ref{['func_comp']})used in Experiments 1, 2, and 4.
• Figure 3: Grid refinement analysis for the numerical integration of the function in (\ref{['function']}). To the left, using the composite trapezoid rule and the corrected composite trapezoid rule. At the center, using the composite Simpson's $1/3$ rule and the corrected rule. To the right, using the composite Simpson's $3/8$ rule and the corrected rule. We can see that the theoretical error decreases with $i$, being $n=2^i$ the number of nodes used. The error of the corrected formulas decreases following the theoretical rate. Tables \ref{['conv']}, \ref{['s_1_3']} and \ref{['s3_8']} contain the data represented in these graphs.
• Figure 4: Grid refinement analysis for the numerical integration of the function in (\ref{['function_corner']}). In this example we use the composite Simpson's $1/3$ rule and the corrected rule. We can see that the error of the corrected formulas decrease following the theoretical rate.
• Figure 5: Representation of the error obtained when choosing randomly 1000 samples for $x^*\in[-1,1]$ and integrating the piece-wise polynomial functions in Table (\ref{['polinomios']}) using the Gaussian quadrature formulas in Table (\ref{['tabla_GL']}), with and without correction terms. In the $x$ axis we have represented the position of the discontinuity $x^*$ and in the $y$ axis, the error obtained with the corresponding Gauss-Legendre quadrature formula. With a circle we have marked the maximum error obtained for the 1000 experiments performed for the corrected and classical formula.

Theorems & Definitions (11)

• Theorem 2.1
• proof
• Corollary 2.2
• proof
• Remark 2.1
• Theorem 3.1
• proof
• Remark 3.1
• Theorem 4.1
• proof