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Application of the Arnoldi Algorithm to the solution of the advection‐dispersion equation
Author(s) -
Woodbury Allan D.,
Dunbar W. Scott,
NourOmid Bahram
Publication year - 1990
Publication title -
water resources research
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.863
H-Index - 217
eISSN - 1944-7973
pISSN - 0043-1397
DOI - 10.1029/wr026i010p02579
Subject(s) - discretization , crank–nicolson method , finite element method , mathematics , matrix (chemical analysis) , algorithm , mathematical optimization , mathematical analysis , materials science , composite material , physics , thermodynamics
The solution to the finite element matrix differential equations resulting from the discretization of the contaminant transport equation is normally carried out by a finite difference approximation to the time derivative. The total computational effort in simulating a contaminant plume is then directly related to the number of unknowns and the number of time steps required to obtain accurate and stable solutions. An alternative method is the Arnoldi algorithm which uses orthogonal matrix transformations to reduce the finite element equations to a much smaller upper Hessenberg system of first‐order differential equations. This new system can be solved by a standard Crank‐Nicolson algorithm with very little computational effort. A matrix‐vector multiplication is then used to obtain the original solution at desired time steps. The algorithm is used to simulate accurately the contaminant plumes for a strip source areal aquifer, a cross‐sectional problem, and the Borden landfill in Ontario, Canada. The Arnoldi algorithm shows an impressive 613% increase in speed over the conventional Crank‐Nicolson scheme for this latter case. The method affords an efficient means of solving large problems, particularly when time durations are long.