Open-File Report 2008–1331
The preconditioned conjugate gradient with improved nonlinear control (PCGN) package provides additional means by which the solution of nonlinear ground-water ﬂow problems can be controlled as compared to existing solver packages for MODFLOW. Picard iteration is used to solve nonlinear ground-water ﬂow equations by iteratively solving a linear approximation of the nonlinear equations. The linear solution is provided by means of the preconditioned conjugate gradient algorithm where preconditioning is provided by the modiﬁed incomplete Cholesky algorithm. The incomplete Cholesky scheme incorporates two levels of ﬁll, 0 and 1, in which the pivots can be modiﬁed so that the row sums of the preconditioning matrix and the original matrix are approximately equal. A relaxation factor is used to implement the modiﬁed pivots, which determines the degree of modiﬁcation allowed. The effects of ﬁll level and degree of pivot modiﬁcation are brieﬂy explored by means of a synthetic, heterogeneous ﬁnite-difference matrix; results are reported in the ﬁnal section of this report. The preconditioned conjugate gradient method is coupled with Picard iteration so as to efﬁciently solve the nonlinear equations associated with many ground-water ﬂow problems. The description of this coupling of the linear solver with Picard iteration is a primary concern of this document.
Posted November 2008
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Naff, R.L., and Banta, E.R., 2008, The U.S. Geological Survey modular ground-water model—PCGN: A preconditioned conjugate gradient solver with improved nonlinear control: U.S. Geological Survey Open-File Report 2008–1331, 35 p.
Description of Picard Iteration Scheme
Input Instructions for the PCGN Solver
Output Diagnostics for the Picard Iteration
Application of PCGN to a Nonlinear Problem
Description of Program Modules
Preconditioned Conjugate Gradient Method
Incomplete Cholesky Preconditioning
Test Results for Modified Incomplete Cholesky Preconditioning