Andrew Lawrence, Department of Applied Mathematics, University of Colorado Boulder

Enhanced trapezoidal rule for discontinuous functions

In many applications, data is available only in the form of function values at equispaced points. For smooth functions, the trapezoidal rule is then highly accurate in periodic cases, but degrades to second order in non-periodic cases. The Gregory approach for end corrections (dating back to 1670) can improve the accuracy order to 9 before the onset of negative weights followed by rapidly increasing ill-conditioning. A recent study showed how this ill-conditioning can be greatly reduced, realizing accuracy orders up through 20 (or even more). In the situation considered in this study, we do not require the end points of the integration interval to coincide with any of the the equispaced grid points. We show here that accuracy orders up to around 10 can be achieved also in this case, with all quadrature weights still remaining non-negative. This method can be utilized for example when integrating piecewise smooth functions, with discontinuities at known in-between grid point locations.