The Problem of Complex Eigensystems in the Semianalytical Solution For Advancement of Time in Solute Transport Simulations: A New Method Using Real Arithmetic

Abstract

In the numerical modeling of groundwater solute transport, explicit solutions may be obtained for the concentration field at any future time without computing concentrations at intermediate times. The spatial variables are discretized and time is left continuous in the governing differential equation. These semianalytical solutions have been presented in the literature and involve the eigensystem of a coefficient matrix. This eigensystem may be complex (i.e., have imaginary components) due to the asymmetry created by the advection term in the governing advection-dispersion equation. Previous investigators have either used complex arithmetic to represent a complex eigensystem or chosen large dispersivity values for which the imaginary components of the complex eigenvalues may be ignored without significant error. It is shown here that the error due to ignoring the imaginary components of complex eigenvalues is large for small dispersivity values. A new algorithm that represents the complex eigensystem by converting it to a real eigensystem is presented. The method requires only real arithmetic.

Semidiscrete pesticide transport modeling and application

“…Then, the ODE system is analytically solved to yield the final closed-form solution. The semidiscrete approach has been applied to contaminant transport modeling in porous media (Guymon, 1970;Guymon et al, 1970;Nalluswami et al, 1972;Willis, 1979;Hwang et al, 1984;Umari and Gorelick, 1986;Kamra et al, 1991a,b;Skaggs and Barry, 1996). This solution method overcomes the limitations to the analytical solution engendered by heterogeneous media and irregular boundary geometry (Hwang et al, 1984).…”

“…Since the solution is continuous and exact in time, no time-marching is necessary for temporal computations, which significantly minimizes numerical error accumulations. Thus, solute concentrations at any spatial point can be calculated without passing through all intermediate time steps (Hwang et al, 1984;Umari and Gorelick, 1986). This technique eliminates artificial smearing and oscillations in the numerical solution induced by time discretization and the relevant schemes, such as implicit or explicit schemes.…”

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

“…These procedures have been also applied by the authors to a solute-transport problem (Umari and Gorelick, 1986). …”

Sensitivity Methods for Time‐Continuous, Spatially Discrete Groundwater Contaminant Transport Models

“…However, they are not suitable for use with simulation equations that employ alternative solution methodologies, such as those that utilize a continuous time domain. Time-continuous solution procedures include the eigenvalue approach [Hwang et al, 1984;Umari and Gorelick, 1986] and various Laplace-transform-based solution techniques. In these latter methods a solution is found in Laplace space and inverted numerically to the time domain.…”