Preconditioned Conjugate-Gradient Package 2 (PCG2)

Last modified on Wednesday, January 31, 1996 - 6:20:49 PM

Skip forward to Input instructions for the Preconditioned Conjugate-Gradient Package 2.

The original instructions for the Preconditioned Conjugate-Gradient Package 2 are on pages 13 to 14 of Hill (1990).

Introduction: The Preconditioned Conjugate-Gradient Package 2 is one of several options for solving the finite difference approximation of the fundamental equations governing groundwater flow in MODFLOW. Other options include the Strongly Implicit Procedure Package and the Slice-Successive Overrelaxation Package. The solutions obtained with any of these solvers should be similar although the time required to find the solution and the amount of memory used may differ. If you wish to use the rewetting capability of BCF2 or BCF3, the Preconditioned Conjugate Gradient Package is usually the best choice. The Slice-Successive Overrelaxation Package is usually inappropriate when the rewetting capability is active. The Strongly Implicit Procedure can be used with rewetting capability although the Preconditioned Conjugate Gradient Package is usually better for that purpose.

Two preconditioning methods may be used; "Modified Incomplete Cholesky", or "Polynomial". The Modified, Incomplete Cholesky method is best for use on scalar computers whereas the Polynomial method is better for vector computers or can be used to save computer storage. (From what I've been able to gather, PC's and Mac's are scalar computers.) Hill (1990) describes how the source code for the Preconditioned Conjugate-Gradient Package 2 can be modified to take better advantage computers with vector and parallel architecture.

There are two different types of iterations in PCG2; outer (Picard) iterations and inner iterations. As their names imply there are several inner iterations within each outer iteration. Cells can not change between wet and dry during the inner iterations.

The number of outer iterations required to reach a solution will vary depending on whether a problem is linear or non-linear. Linear problems will usually only require 1 outer iteration if the number of inner iterations is less than 50. Nonlinear problems are solved most quickly if 3-10 inner iterations occur between each outer iteration. Up to 100 outer iterations may be required. Nonlinearities occur if an aquifer is unconfined or if there is a head-dependent boundary condition which is nonlinear. The following packages have nonlinear head-dependent boundary conditions.

I have not yet determined whether the following packages include nonlinear head-dependent boundary conditions

Transient Leakage Package (TLK)
General Finite Difference Flow Package (GFD)

Unlike other solvers, there are two convergence criteria in PCG2. The first criterion (HCLOSE) is identical to that used in the other solvers; the maximum change in head between successive iterations must be smaller than some user-specified amount. The other criterion is unique to the PCG solvers. As I understand it, it is related to the process whereby PCG2 solves the finite difference flow equations. PCG2 takes the flow equations and changes them into a form which can be easily solved. However, the flows computed with the modified equations are not identical to those computed with the original equations. The second criterion is satisfied if the maximum difference between the flow computed with the original and modified equations between any two cells is less than RCLOSE. Both criteria must be satisfied simultaneously.

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Input Instructions

All input parameters should be right justified.

Line 1
- Spaces 1-10, MXITER, Integer, Maximum number of outer iterations per time step. Usually 1 for linear problems. Usually <=100 for nonlinear problems.
- Spaces 10-20, ITER1, Integer, Maximum number of inner iterations, Usually <=30 for linear problems, usually 3-10 for nonlinear problems.
- Spaces 21-30, NPCOND, Integer,
  - If NPCOND=1, the Modified, Incomplete Cholesky method will be used. (scalar computers)
  - If NPCOND=2, the Polynomial method will be used. (vector computers or to save computer storage)
Line 2
- Spaces 1-10, HCLOSE, Real Number, Head change criterion for convergence
- Spaces 11-20, RCLOSE, Real Number, Residual criterion for convergence
- Spaces 21-30, RELAX, Real Number, Relaxation parameter, usually set to 1, used only if NPCOND=1.
- Spaces 31-40, NBPOL, Integer, If NBPOL=2, the estimate of the upper bound on the maximum eigenvalue is 2, otherwise the estimate will be calculated, used only if NPCOND=2.
- Spaces 41-50, IPRPCG, Integer, Printout interval , if set equal to 0 it will be changed to 999
- Spaces 51-60, MUTPCG, Integer,
  - if MUTPCG <1, number of iterations, head changes and residuals are printed
  - if MUTPCG =1, only the number of iterations is printed
  - if MUTPCG >1, all printing is suppressed.
- Spaces 61-70, IPCGCD, Integer, 0 for most applications, used only if NPCOND=1

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Line 1, Spaces 1-10, MXITER, Integer

Maximum number of outer iterations per time step. For linear problems, MXITER, should be 1 unless more than 50 inner iterations are required in which case MXITER could be as large as 10. For nonlinear problems, MXITER may need to be larger but should usually be less than 100.

On each time step MODFLOW will go through a series of iterations as it approaches an acceptable solution. Sometimes, however, the program may enter an infinite loop. To prevent this, MXITER forces the program to terminate after a number of iterations. If this happens, MODFLOW has not reached a solution and the results should not be accepted as a good estimate of the groundwater flow. Instead the input needs to be modified to avoid this problem. Often the problem is with the conceptual model. You should go back and check that your conceptual model is reasonable. Drying and rewetting of cells with the BCF2 or BCF3 packages is also a frequent culprit.