Solving the Eigenvalue problem for a covariance kernal with variable correlation length

Abstract

In stochastic modelling of flow in porous media, the medium properties that produce random trajectories of fluid elements are modelled by the assumed correlation kernel. For numerical simulation of the flow, the Stochastic Differential Equation (SDE) is expanded using a Karhunen-Loeve expansion in terms of Eigenvalues and Eigenfunctions of the correlation function. In real world problems such as contaminant transport in aquifers, the medium properties are themselves variable necessitating the solution of the eigenvalue integral equation with a variable correlation length. We investigate this problem by comparing several approximate approaches for an assumed 1-dimensional exponential kernel with variable correlation length λ. In matrix methods, the known solutions of the fixed λ equation for a representative value λ⁰ , are used to expand the variable λ solutions and hence convert the integral equation to a matrix eigenvalue equation. It is shown that calculation of the matrix is the computational bottleneck and two approximations are introduced that speeds this up by two orders of magnitude. One of these is to use a piecewise constant kernel; this leads also to a non-matrix method where the Eigenfunction itself is approximated as a piecewise function. The performance of these approximations is investigated in detail by applying them to a model problem and it is found that both the best speed and accuracy is achieved by a method that uses the fact that the correlation function is strongly localised. It is shown that this approximation, called the diagonal correlation length matrix method, gives virtually identical eigenvalues and eigenfuntions to an exact calculation of matrix elements. Its use to give a basis for expanding stochastic quantities is illustrated by showing an expansion of the covariance function in terms of the variable λ Eigenfunctions.