Accurate integration rules for functions with singularities

This work is devoted to the construction and analysis of a new nonlinear technique that allows to improve the accuracy of classical numerical integration formulas of any order when dealing with data that contains discontinuities. The novelty of the technique consists in the inclusion of cor- rection terms with a closed expression that depends on the size of the jumps of the function and its derivatives at the discontinuities. The addition of these terms allows to recover the accuracy of classical numerical integration formulas even 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 a post-processing, which is useful if the main calculation of the integral has been already done using classical formulas. The numerical experiments performed allow to confirm the theoretical conclusions reached in this paper.