Variational analysis for simulating free-surface flows in a porous medium

A variational formulation has been developed to solve a parabolic partial differential equation describing free-surface flows in a porous medium. The variational finite element method is used to obtain a discrete form of equations for a two-dimensional domain. The matrix characteristics and the sta...

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Main Authors: Shabbir Ahmed, Charles Collins
Format: Article
Language:English
Published: Wiley 2003-01-01
Series:Journal of Applied Mathematics
Online Access:http://dx.doi.org/10.1155/S1110757X03301147
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author Shabbir Ahmed
Charles Collins
author_facet Shabbir Ahmed
Charles Collins
author_sort Shabbir Ahmed
collection DOAJ
description A variational formulation has been developed to solve a parabolic partial differential equation describing free-surface flows in a porous medium. The variational finite element method is used to obtain a discrete form of equations for a two-dimensional domain. The matrix characteristics and the stability criteria have been investigated to develop a stable numerical algorithm for solving the governing equation. A computer programme has been written to solve a symmetric positive definite system obtained from the variational finite element analysis. The system of equations is solved using the conjugate gradient method. The solution generates time-varying hydraulic heads in the subsurface. The interfacing free surface between the unsaturated and saturated zones in the variably saturated domain is located, based on the computed hydraulic heads. Example problems are investigated. The finite element solutions are compared with the exact solutions for the example problems. The numerical characteristics of the finite element solution method are also investigated using the example problems.
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publishDate 2003-01-01
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record_format Article
series Journal of Applied Mathematics
spelling doaj-art-5675fdf33e6442bfae7ff6f46e631b5e2025-02-03T05:45:10ZengWileyJournal of Applied Mathematics1110-757X1687-00422003-01-012003837739610.1155/S1110757X03301147Variational analysis for simulating free-surface flows in a porous mediumShabbir Ahmed0Charles Collins1Big Cypress Basin, South Florida Water Management District, 6089 Janes Lane, Naples 34109, FL, USADepartment of Mathematics, University of Tennessee, 121 Ayres Hall, 1403 Circle Drive, Knoxville 37996, TN, USAA variational formulation has been developed to solve a parabolic partial differential equation describing free-surface flows in a porous medium. The variational finite element method is used to obtain a discrete form of equations for a two-dimensional domain. The matrix characteristics and the stability criteria have been investigated to develop a stable numerical algorithm for solving the governing equation. A computer programme has been written to solve a symmetric positive definite system obtained from the variational finite element analysis. The system of equations is solved using the conjugate gradient method. The solution generates time-varying hydraulic heads in the subsurface. The interfacing free surface between the unsaturated and saturated zones in the variably saturated domain is located, based on the computed hydraulic heads. Example problems are investigated. The finite element solutions are compared with the exact solutions for the example problems. The numerical characteristics of the finite element solution method are also investigated using the example problems.http://dx.doi.org/10.1155/S1110757X03301147
spellingShingle Shabbir Ahmed
Charles Collins
Variational analysis for simulating free-surface flows in a porous medium
Journal of Applied Mathematics
title Variational analysis for simulating free-surface flows in a porous medium
title_full Variational analysis for simulating free-surface flows in a porous medium
title_fullStr Variational analysis for simulating free-surface flows in a porous medium
title_full_unstemmed Variational analysis for simulating free-surface flows in a porous medium
title_short Variational analysis for simulating free-surface flows in a porous medium
title_sort variational analysis for simulating free surface flows in a porous medium
url http://dx.doi.org/10.1155/S1110757X03301147
work_keys_str_mv AT shabbirahmed variationalanalysisforsimulatingfreesurfaceflowsinaporousmedium
AT charlescollins variationalanalysisforsimulatingfreesurfaceflowsinaporousmedium