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PCGN: A Preconditioned Conjugate Gradient Solver with Improved Nonlinear Control


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Summary:

The preconditioned conjugate gradient solver with improved nonlinear control (PCGN) is a new solver package for the MODFLOW ground-water flow model (Harbaugh and others, 2000). The principal objective of the PCGN package is to provide the modeler with more options when faced with a poorly converging nonlinear problem. In MODFLOW, nonlinear problems are solved by iteratively solving a linearized approximation of the problem. Because MODFLOW uses a cell-centered finite-difference (CCFD) approximation of the ground-water flow equations, the linear approximation consists of a system of equations represented by a sparse, regular matrix. The linear equation solver in the PCGN package is based in the preconditioned conjugate gradient (PCG) algorithm; preconditioning is provided by means of the the incomplete Cholesky algorithm with two fill-level options: 0 and 1. For a fill level of 0, the incomplete Cholesky algorithm decomposition does not allow for factors outside the existing nonzero entries in the CCFD matrix; this fill level is generally accepted as the traditional incomplete Cholesky preconditioning (van der Vorst, 2003). The fill level 1 is associated with the next higher-order factorization; additional Cholesky factors are formed and stored for this fill level.

The iterative procedure used in MODFLOW for solving nonlinear problems is commonly referred to as Picard iteration. Within the PCGN package, the principal controls on the Picard iteration are the convergence parameter for obtaining a head-change solution from the PCG solver and the damping factor for updating the nonlinear solution. The accuracy of the updated head change is determined by the convergence parameter, while the damping factor dictates the proportion of the updated head change to be added to the nonlinear solution. The PCGN package gives the modeler some additional tools to manipulate these parameters within the context of the Picard iteration. In particular, both the damping factor and convergence parameter can be made adaptive in that their values are made to depend on the progress of the nonlinear iteration. In addition, when adaptive damping is elected, it is also feasible to limit the maximum head change applied in any given Picard iteration. Limiting the maximum head is useful when modeling dewatering scenarios as the linear approximation can produce very large and abrupt head changes. Finally, options also exist to institute smaller values for the damping factor and convergence parameter initially in each new stress period, but relax those values with favorable progress in the nonlinear iteration.

References:

Harbaugh, A., Banta, E., Hill, M., and McDonald, M., 2000, Modflow-2000, the U.S. Geological Survey modular ground-water model--User guide to modularization concepts and the ground-water flow process: U.S. Geological Survey Open-File Report 00--92, 121 p.

van der Vorst, H., 2003, Iterative Krylov methods for large linear systems: Cambridge, U.K., Cambridge University Press, 221 p.

Availability:

The PCGN solver is made available in source form for either MODFLOW-2000 or MODFLOW-2005 and MODFLOW-LGR (MODFLOW-2005 and MODFLOW-LGR versions of PCGN are interchangeable). A Makefile, appropriate to compiling MODFLOW with PCGN on Linux and Unix platforms, is bundled with the source files so as to assist Linux and Unix users. Most Linux platforms come with gfortran installed (or readily available), which makes installation from the source files a relatively simple task. Precompiled executable versions of MODFLOW-2000, MODFLOW-2005 and MODFLOW-LGR with the PCGN solver for installation on Microsoft Windows platforms are also made available to Windows users as appropriate compilers are not as common for these platforms.

NOTICE!!!

The author of PCGN is considering removing adaptive damping option 2 (ADAMP=2) and adaptive convergence option 1 (ACNVG=1) from PCGN; if anyone is using these options, please notify the author ASAP.

Update history:

bullet April 15, 2009: Repaired indexing problem in PCGN associated with reporting location of maximum head change and zero pivots; repair does not affect operation of solver.
bullet December 10, 2009: Updated PCGN package to work with version 1.7 of MODFLOW-2005 (modifications to MODFLOW main and Makefile). Corrected error in PCGN usage of subroutine gwf2mnw7.f in MODFLOW-2005. No changes to MODFLOW-2000 or MODFLOW-LGR.
bullet Feburary 25, 2010: Corrected problem in PCGN associated with terminating simulation when number of outer iterations (KITER) exceeds allowed maximum (ITER_MO): Simulation was not terminating in this event. Problem affected MODFLOW-2005 but not MODFLOW-2000.
bullet November 23, 2010: Updated PCGN package to work with version 1.8 of MODFLOW-2005 (modifications to MODFLOW main, the Makefile and files gwf2mnw17.f and gwf2mnw27.f; no changes to the PCGN package).

Instructions and Downloads:

PCGN documentation and a test problem are provided with all source-code and executable releases.

bullet INSTRUCTIONS: Installing MODFLOW-2000 and MODFLOW-2005 from source code on Linux/Unix operating systems
bullet INSTRUCTIONS: Installing MODFLOW-LGR from source code on Linux/Unix operating systems
bullet INSTRUCTIONS: Installing MODFLOW-2000 and MODFLOW-2005 from precompiled executable on Microsoft Windows operating systems
bullet INSTRUCTIONS: Installing MODFLOW-LGR from precompiled executable on Microsoft Windows operating systems
bullet DOWNLOAD: PCGN release for MODFLOW-2000; source code, requires Fortran and C compilers, zipped tar file (0.5MB)
bullet DOWNLOAD: PCGN release for MODFLOW-2005: source code, requires Fortran and C compilers, zipped tar file (0.5MB)
bullet DOWNLOAD: MODFLOW-LGR with PCGN: source code for version 1.1.1, requires Fortran and C compilers, zipped tar file (1.8MB)
bullet DOWNLOAD: Microsoft Windows Executable for MODFLOW-2000: version 1_18.01, zip file (1.6MB)
bullet DOWNLOAD: Microsoft Windows Executable for MODFLOW-2005: version 1.7.00, zip file (1.7MB)
bullet DOWNLOAD: Microsoft Windows Executable for MODFLOW-LGR: version 1.1.1, zip file (2.7MB)
bullet DOWNLOAD: Documentation (Version 1) pdf file (0.8MB)

Note: To view this pdf file, you need the Adobe Acrobat® Reader installed on your computer.
A free copy of the Acrobat® Reader may be downloaded from Adobe Systems Incorporated.

Descriptions:

bullet PCGN (Version 1): Abstract
bullet Consult the official U.S.G.S. MODFLOW webpage here .

Communication:

bullet Send in Bug Reports/Help Requests/Comments
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DISCLAIMER STATEMENTS

Although this program has been used by the USGS, NO WARRANTY, expressed or implied, is made by the USGS or the United States Government as to the accuracy and functioning of the program and related program material nor shall the fact of distribution constitute any such warranty, and no responsibility is assumed by the USGS in connection therewith.

Please note that some U.S. Geological Survey (USGS) information accessed through this page may be preliminary in nature and presented prior to final review and approval by the Director of the USGS. This information is provided with the understanding that it is not guaranteed to be correct or complete and conclusions drawn from such information are the sole responsibility of the user.

Any references to commercial vendors of software products or services are provided solely for the convenience of users when obtaining or using USGS software. Such references do not imply any endorsement by the U.S. Government.


URL: http://wwwbrr.cr.usgs.gov/projects/GW_stoch/pcgn/index.shtml
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Last Modified: 23 Nov 2010
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