Skip Links

USGS - science for a changing world

Open-File Report 2008–1331

The U.S. Geological Survey Modular Ground-Water Model—PCGN: A Preconditioned Conjugate Gradient Solver with Improved Nonlinear Control

By Richard L. Naff and Edward R. Banta


Thumbnail of and link to report PDF (776 kB)

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.

Version 1.0

Posted November 2008

For additional information contact:

Chief, Branch of Regional Research,
Central Region Box 25046,
Mail Stop 418
Denver, CO 80225

Or visit the USGS National Research Program Web site at: http:/

Part or all of this report is presented in Portable Document Format (PDF); the latest version of Adobe Reader or similar software is required to view it. Download the latest version of Adobe Reader, free of charge.

Suggested citation:

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

References Cited

Accessibility FOIA Privacy Policies and Notices

Take Pride in America logo logo U.S. Department of the Interior | U.S. Geological Survey
Page Contact Information: Contact USGS
Page Last Modified: Wednesday, December 07 2016, 09:35:16 PM