Numerical integration rules based on B-spline bases
Dionisio F. Yáñez
TL;DR
This work develops high-order numerical integration rules by perturbing the trapezoidal rule with B-spline-based quasi-interpolation, enabling arbitrary order accuracy $p$ and a tensor-product extension to multiple dimensions. Central to the approach is a quasi-interpolation operator $Q_p$ that reproduces polynomials up to degree $p$ and a relation between cell-average data and point values via B-splines, leading to both one-subinterval and composite rules characterized by B-integration weights $\tau_{p,j}$ and cumulative weights $\xi_{p,i}$. The authors connect these rules to Euler–Maclaurin corrections and show that, with appropriate derivative approximations, the new formulas align with known high-order schemes, while offering practical formulations with controlled evaluation costs. Numerical experiments on $f_1(x)=f_1(x)=\$f_1(x)=e^{x^2}$ and Runge-type functions demonstrate the expected convergence orders and highlight potential efficiency gains relative to trapezoidal and Simpson rules, supporting the method as a flexible framework for high-order, cost-aware integration.
Abstract
In this work, we present some new integration formulas for any order of accuracy as an application of the B-spline relations obtained in [1]. The resulting rules are defined as a perturbation of the trapezoidal integration method. We prove the order of approximation and extend the results to several dimensions. Finally, some numerical experiments are performed in order to check the theoretical results.