# Generalized alternating‐direction collocation methods for parabolic equations. II. Transport equations with application to seawater intrusion problems

Michael Anthony Celia, George F. Pinder

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12 Scopus citations

## Abstract

The alternating‐direction collocation (ADC) method combines the attractive computational features of a collocation spatial approximation and an alternating‐direction time marching algorithm. The result is a very efficient solution procedure for parabolic partial differential equations. To date, the methodology has been formulated and demonstrated for second‐order parabolic equations with insignificant first‐order derivatives. However, when solving transport equations, significant first‐order advection components are likely to be present. Therefore, in this paper, the ADC method is formulated and analyzed for the transport equation. The presence of first‐order spatial derivatives leads to restrictions that are not present when only second‐order derivatives appear in the governing equation. However, the method still appears to be applicable to a wide variety of transport systems. A formulation of the ADC algorithm for the nonlinear system of equations that describes density‐dependent fluid flow and solute transport in porous media demonstrates this point. An example of seawater intrusion into coastal aquifers is solved to illustrate the applicability of the method. An alternating‐direction collocation solution algorithm has been developed for the general transport equation. The procedure is analogous to that for the model parabolic equations considered by Celia and Pinder . However, the presence of first‐order spatial derivatives requires special attention in the ADC formulation and application. With proper implementation, the ADC procedure effectively combines the efficient equation formulation inherent in the collocation method with the efficient equation solving characteristics of alternating‐direction time marching algorithms. To demonstrate the viability of the method for problems with complex velocity fields, the procedure was applied to the problem of density‐dependent flow and contaminant transport in groundwaters. A standard example of seawater intrusion into coastal aquifers was solved to illustrate the applicability of the method and to demonstrate its potential use in practical problems.

Original language English (US) 215-230 16 Numerical Methods for Partial Differential Equations 6 3 https://doi.org/10.1002/num.1690060303 Published - Jan 1 1990

## All Science Journal Classification (ASJC) codes

• Analysis
• Numerical Analysis
• Computational Mathematics
• Applied Mathematics