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Mathematical Problems in Engineering
Volume 2016, Article ID 7860618, 15 pages
http://dx.doi.org/10.1155/2016/7860618
Research Article

Optimized Finite Difference Formulas for Accurate High Frequency Components

Universidad Carlos III de Madrid, Madrid, Spain

Received 8 August 2016; Accepted 9 November 2016

Academic Editor: Xesús Nogueira

Copyright © 2016 Manuel Kindelan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We present a method to obtain optimal finite difference formulas which maximize their frequency range of validity. The optimization is based on the idea of keeping an error of interest (dispersion, phase, or group velocities errors) below a given threshold for a wavenumber interval as large as possible. To find the weights of these optimal finite difference formulas we solve a system of nonlinear equations. Furthermore, we give compact formulas for the optimal weights as function of the error bound. Several numerical experiments illustrate the performance of the obtained finite difference formulas compared to the standard ones.