Improving the accuracy of the trapezoidal rule
The trapezoidal rule uses function values at equispaced nodes. It is very accurate for integrals over periodic intervals, but is usually quite inaccurate in non-periodic cases. Commonly used improvements, such as Simpson’s rule and the Newton-Cotes formulas, are not much (if at all) better than the even more classical quadrature method by James Gregory (1638-1675). For increasing orders of accuracy, these methods all suffer from the Runge phenomenon (the fact that polynomial interpolants on equispaced grids become violently oscillatory as their degree increases). In the context of quadrature methods, and for orders of accuracy around 10 or higher, the Runge phenomenon leads to weights of oscillating signs and large magnitudes.
When looking further into a recently developed (radial basis function-based) method for numerical quadrature over curved bounded surfaces, we noted that this approach somehow managed avoid adverse Runge phenomenon-type effects. This inspired the method that will be focused on here – an enhancement to Gregory’s method that can be of very high order of accuracy without any weights becoming large in magnitude (still on equispaced grids).
The present work has been carried out in collaboration with Prof. Jonah Reeger (US Naval Academy, Annapolis, Maryland; Ph.D. from CU APPM in 2013).