Finite Analytic Numerical Solution for Two‐Dimensional Groundwater Solute Transport

Abstract

A new numerical method called the finite analytic (FA) method is used to solve a groundwater solute transport problem. The basic idea of the finite analytic method is the incorporation of local analytic solution in the numerical solution of the partial differential equation. When the local analytic solution is evaluated at a given nodal point, it gives an algebraic relationship between a nodal value in an element and its neighboring nodal points. The assemble of the linear system equations results in a tridiagonal matrix. Like most finite difference method, the advantages of using efficient iterative techniques for solving tridiagonal matrices are equally applicable to FA method. The automatic localized upstream shift and the analytic property of the FA method eliminates the difficulty of numerical dispersion locally and suppresses the overall numerical dispersion for large Peclet number. For small Peclet number FA method yields excellent results in comparison with the analytic solution. For large Peclet number FA solutions are oscillation free with some degree of numerical dispersion. The results are comparable with those obtained using upstream weighted finite element method. INTRODUCTION This paper employs a new numerical method called the finite analytic (FA) method to study the two-dimensional solute transport in groundwater. The FA method was introduced by Chen and Li [1979] and Chen et al. [1981] to solve heat transfer problems. The method is neither a finite difference (FD) method nor a finite element (FE) method. The basic idea of the FA method is to invoke the analytic solution of the governing differential equation in the numerical solution of the problem. In the FA method the total problem is subdivided into small elements. The local analytical solution is then expressed in an algebraic form, relating an interior nodal value to its neighboring nodal values. The system of algebraic equations derived from the local solution is then solved to provide the solution of the total problem. Satisfactory numerical solution of groundwater solute transport equation can generally be obtained for small Peclet number (Pe = VL/D). However, for large Pe (e.g., small D) the numerical solutions characteristically exhibit either oscillations in concentration in the neighborhood of a sharp front or a smearing of the front. Thus numerical methods are often evaluated in terms of frontal smearing and concentration overshoot in the numerical solutions.Gray and Pinder [1976] assumed that the spatial distribution of concentration can be expressed as a Fourier series with components of different wavelengths. When the numerical dissipation (numerically induced errors that arise in computing the amplitude of the Fourier components) is large, the small wavelengths important to describing a sharp concentration front are damped out and the front is smeared. Large numerical dispersion (errors in a numerical solution caused by propagating a Fourier component at an incorrect speed) causes small wavelength to be out of phase with the la...

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