Frequency optimized computation methods

In this paper we develop an alternative method to derive finite difference approximations of derivatives on arbitrary distrubutions of data points. The purpose is to find schemes which work for a broader range of frequencies than the usual approximations based on polynomial fitting to the expense of less accuracy for low frequencies. The numerical schemes are obtained as solutions to constrained optimizations problems in a weighted L-2-norm in the frequency domain. We examine the accuracy of these schemes and compare them with the standard approximations. To test the accuracy of the different schemes, we study dispersion errors for a simple wave equation in one space dimension. We examine the number of points per wave length which is needed in order for the relative error in the phase velocity to be below a certain bound. We also apply the technique to solve a simple two-dimensional hyperbolic equation.