Analysis of Spatial Variability Using a Numerical Spectral Approach

Abstract

Numerical spectral methods are alternatives to finite elements and finite differences for solving partial differential equations. In this paper, we discuss how characteristics particular to spectral methods make them ideally suited for some problems of relating covariances using the physically based governing equations. We examine the approximations required for the numerical method, and compare numerical spectral methods to finite element and finite difference methods. Finally, we demonstrate how the numerical spectral method can be applied to any valid log-permeability covariance by providing an example using a piece-wise linear covariance. The numerical spectral methods are computationally efficient and flexible. We conclude that numerical spectral methods offer an excellent practical way of using stochastic partial differential equations to derive the statistical characteristics of variables of interest in geohydrologic applications.