Numerical integration rules with improved accuracy close to discontinuities

Abstract

This work is devoted to the construction and analysis of a new nonlinear technique that allows obtaining accurate numerical integrations of any order using data that contains discontinuities, and when the integrand is only known at grid points. The novelty of the technique consists in the inclusion of correction terms with a closed expression that depend on the size of the jumps of the function and its derivatives at the discontinuities, that are supposed to be known. The addition of these terms allows recovering the accuracy of classical numerical integration formulas close to the discontinuities, as these correction terms account for the error that the classical integration formulas commit up to their accuracy at smooth zones. Thus, the correction terms can be added during the integration or as post-processing, which is useful if the main calculation of the integral has been already done using classical formulas. We include several numerical experiments that confirm the theoretical conclusions reached in this article.