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Evaluation of the matrix exponential for use in ground-water-flow and solute-transport simulations; theoretical framework

Water-Resources Investigations Report 86-4096
By:  and 
https://doi.org/10.3133/wri864096

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Abstract

It is possible to obtain analytic solutions to the groundwater flow and solute transport equations if space variables are discretized but time is left continuous. From these solutions, hydraulic head and concentration fields for any future time can be obtained without ' marching ' through intermediate time steps. This analytical approach involves matrix exponentiation and is referred to as the Matrix Exponential Time Advancement (META) method. Two algorithms are presented for the META method, one for symmetric and the other for non-symmetric exponent matrices. A numerical accuracy indicator, referred to as the matrix condition number, was defined and used to determine the maximum number of significant figures that may be lost in the META method computations. The relative computational and storage requirements of the META method with respect to the time marching method increase with the number of nodes in the discretized problem. The potential greater accuracy of the META method and the associated greater reliability through use of the matrix condition number have to be weighed against this increased relative computational and storage requirements of this approach as the number of nodes becomes large. For a particular number of nodes, the META method may be computationally more efficient than the time-marching method, depending on the size of time steps used in the latter. A numerical example illustrates application of the META method to a sample ground-water-flow problem. (Author 's abstract)

Suggested Citation

Umari, A., and Gorelick, S., 1986, Evaluation of the matrix exponential for use in ground-water-flow and solute-transport simulations; theoretical framework: U.S. Geological Survey Water-Resources Investigations Report 86-4096, iv, 33 p. :ill. ;28 cm., https://doi.org/10.3133/wri864096.

Publication type Report
Publication Subtype USGS Numbered Series
Title Evaluation of the matrix exponential for use in ground-water-flow and solute-transport simulations; theoretical framework
Series title Water-Resources Investigations Report
Series number 86-4096
DOI 10.3133/wri864096
Year Published 1986
Language ENGLISH
Publisher U.S. Geological Survey,
Description iv, 33 p. :ill. ;28 cm.
Additional publication details