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