A computationally efficient, yet relatively simple Eulerian-Lagrangian method is proposed for solving the one-dimensional convection-dispersion solute transport equation assuming a steady or transient velocity field. The method uses a modified single-step reverse particle tracking (MSRPT) technique to handle steep concentration fronts. The scheme utilizes two weighting factors to control the movement of particles during a backward tracking step. One weighting factor greater than unity is used in the upstream region of the convection front, while another weighting factor less than unity is taken in the downstream region. The two factors were related empirically to the grid Peclet and Courant numbers. The MSRPT technique is carried out only within the concentration plume at each time step. For transient flow fields, the weighting factors were determined using an automatically adjustable procedure based on mass balance errors. The MSRPT method maintains the advantages of the traditional single-step reverse particle tracking (SRPT) procedure, i.e., producing efficient and oscillation-free calculations, but circumvents numerical dispersion introduced by SRPT. A large number of tests against analytical solutions for one-dimensional transport in uniform flow fields indicate that the proposed method can handle the entire range of Peclet numbers from zero to infinity. Numerical tests also show that the MSRPT method is a relatively accurate, efficient and massconservative algorithm for solute transport in transient flow fields. The Courant number at present cannot exceed 1. The MSRPT approach was found especially useful for convection-dominated problems; in fact, an exact numerical solution may be obtained with MSRPT for pure convection.Convection-dispersion type equations are being widely used to model solute transport in soil and groundwater. Owing to the particular combination of hyperbolic and parabolic terms, serious difficulties are often encountered in obtaining accurate numerical solutions of these equations. A variety of numerical schemes have been developed to deal with these difficulties, including an extensive number of Eulerian methods using fixed grid systems, and Lagrangian approaches involving moving coordinates. There is now a growing trend to combine the Eulerian and Lagrangian methods [Garder et al., 1964;Leith, 1965; Heinrich et al., 1977;Konikow and Bredehoeft, 1978;Varoglu and Finn, 1978;Cheng et al., 1984;Khaleel and Reddell, 1985;Fujinawa, 1986;Dimou and Adams, 1991;Leonard, 1991;Galeati et al., 1992;Huang et al., 1992, among others]. Eulerian-Lagrangian methods treat the convection part of the transport problem using a Lagrangian formulation in a fixed Eulerian grid, while the dispersion term is treated by either an Eulerian or a Lagrangian formulation. As a typical example, Neuman and Sorek [1982] decomposed the concentration field into two parts, one controlled by pure convection, the other by dispersion. The convection part was handled with the help of moving particles, while the dispersion p...
A relatively simple method of characteristics is developed to simulate one‐dimensional variably saturated water flow. The method uses the Eulerian‐Lagrangian approach to separate the governing flow equation into “convection” and “diffusion” parts, which are solved with the method of characteristics and the conventional finite element method, respectively. The method of characteristics combines a single‐step reverse particle tracking technique with a correction strategy to ensure accurate mass balances. The correction process is implemented by weighing the calculated convective contribution to the pressure head at each node with the pressure head values of two upstream nodes, using an adaptive weighing factor λ. The value of λ is automatically adjusted by considering the global mass balance at each time step. Numerical experiments for ponded infiltration are presented to illustrate the scheme's performance for situations involving highly nonlinear soil hydraulic properties and extremely dry initial conditions. Results indicate that the proposed method is mass‐conservative, virtually oscillation‐free, and computationally quite efficient. The method is especially effective for simulating highly nonlinear flow scenarios for which traditional finite difference and finite element numerical methods often fail to converge.
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