Computational aspects of the choice of operator and sampling interval for numerical differentiation in large-scale simulation of wave phenomena*

TLDR: In this article, a generalized numerical dispersion analysis for wave equation computations is developed, which can then be designed by minimizing the corresponding peak relative error in group velocity within a spatial frequency band.

Abstract: Conventional finite-difference operators for numerical differentiation become progressively inaccurate at higher frequencies and therefore require very fine computational grids. This problem is avoided when the derivatives are computed by multiplication in the Fourier domain. However, because matrix transpositions are involved, efficient application of this method is restricted to computational environments where the complete data volume required by each computational step can be kept in random access memory. To circumvent these problems a generalized numerical dispersion analysis for wave equation computations is developed. Operators for spatial differentiation can then be designed by minimizing the corresponding peak relative error in group velocity within a spatial frequency band. For specified levels of maximum relative error in group velocity ranging from 0.03% to 3%, differentiators have been designed that have the largest possible bandwidth for a given operator length. The relation between operator length and the required number of grid points per shortest wavelength, for a required accuracy, provides a useful starting point for the design of cost-effective numerical schemes. To illustrate this, different alternatives for numerical simulation of the time evolution of acoustic waves in three-dimensional inhomogeneous media are investigated. It is demonstrated that algorithms can be implemented that require fewer arithmetic and I/O operations by orders of magnitude compared to conventional second-order finite-difference schemes to yield results with a specified minimum accuracy.