Advection-dispersion equation numerical solution

The numerical solution to the advection-dispersion equation and associated adsorption equations can be performed using finite difference schemes, either in their implicit and/or explicit form. In the one-dimensional MRTM model (Selim et al., 1990), the Crank-Nicholson algorithm was applied to solve the governing equations of the chemical transport and retention in soils. The web-based simulation system for the one-dimensional MRTM model is detailed in Zeng et al. (2002). The alternating direction-implicit (ADI) method is used here to solve the three-dimensional models. 

To understand the behavior of the movement of the contaminant in ground-water, people solve Eq. (1) forward in time. In solving this equation forward in time, one assumes that the plume is originated from somewhere and will travel through the porous media due to advection and dispersion. The conventional procedure to solve Eq. (1) is to use finite difference or finite element methods. For simple cases, closed-form solutions exist. Quantitative descriptions of the processes forward in time are well understood. Multidimensional models of these processes have been used successfully in practice [50]. Numerical solute transport models were first developed about 25 years ago. When properly applied, these models can provide useful information about transport processes and can assist in the design of remedial programs. 

The coupled set of nonlinear differential equations (equations 1 and 4) are solved by the alternating direction implicit (ADI) method f9-10) on an evenly spaced grid. The advective transport of a solute species was solved using the Lax-Wendroff two-step method (10). To ensure that numerical dispersion is avoided, a grid spacing was chosen such that the grid Peclet number (defined by < 2 fll). The computational expense involved in using a... 

The truncation error associated with convection/advection schemes can be analyzed by using the modified equation method [254]. By use of Taylor series all the time derivatives except the 1. order one are replaced by space derivatives. When the modified equation is compared with the basic advection equation, the right-hand side can be recognized as the error. The presence of Ax in the leading error term indicate the order of accuracy of the scheme. The even-ordered derivatives in the error represent the diffusion error, while the odd-ordered derivatives represent the dispersion (or phase speed) error. Another method for analyzing the truncation error of advection schemes is the Fourier (or von Neumann method) [157, 158, 215]. This method is used to study the effects of numerical diffusion on the solution. 

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