Premium
A Fast Direct Solution Method for Nonlinear Equations in an Analytic Element Model
Author(s) -
Haitjema Henk M.
Publication year - 2005
Publication title -
groundwater
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.84
H-Index - 94
eISSN - 1745-6584
pISSN - 0017-467X
DOI - 10.1111/j.1745-6584.2005.00145.x
Subject(s) - coefficient matrix , gaussian elimination , mathematics , nonlinear system , boundary element method , matrix (chemical analysis) , iterative method , system of linear equations , linear equation , lu decomposition , independent equation , simultaneous equations , finite element method , mathematical analysis , mathematical optimization , matrix decomposition , differential equation , gaussian , physics , eigenvalues and eigenvectors , materials science , quantum mechanics , composite material , thermodynamics
The analytic element method, like the boundary integral equation method, gives rise to a system of equations with a fully populated coefficient matrix. For simple problems, these systems of equations are linear, and a direct solution method, such as Gauss elimination, offers the most efficient solution strategy. However, more realistic models of regional ground water flow involve nonlinear equations, particularly when including surface water and ground water interactions. The problem may still be solved by use of Gauss elimination, but it requires an iterative procedure with a reconstruction and decomposition of the coefficient matrix at every iteration step. The nonlinearities manifest themselves as changes in individual matrix coefficients and the elimination (or reintroduction) of several equations between one iteration and the other. The repeated matrix reconstruction and decomposition is computationally intense and may be avoided by use of the Sherman‐Morrison formula, which can be used to modify the original solution in accordance with (small) changes in the coefficient matrix. The computational efficiency of the Sherman‐Morrison formula decreases with increasing numbers of equations to be modified. In view of this, the Sherman‐Morrison formula is only used to remove equations from the original set of equations, while treating all other nonlinearities by use of an iterative refinement procedure.