High-Order Finite Differences and the Pseudospectral Method on Staggered Grids

Finite-difference approximations for the first derivative, valid halfway between equidistant gridpoints, are in general much more accurate than the corresponding approximations, which are valid at gridpoints. The pseudospectral (Fourier) method can be viewed as the limit of finite-difference approximations when the order of accuracy tends to infinity. A fundamentally different (and more accurate) pseudospectral method is obtained from the “halfway” approximations. This study derives these and related methods and discusses their accuracies.