SUM-ACCELERATED PSEUDOSPECTRAL METHODS - FINITE-DIFFERENCES AND SECH-WEIGHTED DIFFERENCES

This work continues our previous studies of algorithms for accelerating the convergence of pseudospectral derivative series in order to obtain new differentiation schemes which have the sparsity and low cost of finite differences but the accuracy of spectral methods; We develop a general theoretical framework for difference schemes. Finite differences are close to optimum, but can be bettered by a new scheme we have dubbed 'sech-weighted' differences. Through numerical examples, we show that sech-weighted differences are effective. In contrast, non-linear accelerations like Pade approximants and the Levin u-transform, so popular in other applications, are inferior for approximating derivatives to linear accelerations like finite differences, sech-weighted differences and the Euler method.