UNITED STATES DEPARTMENT OF THE INTERIOR GEOLOGICAL SURVEY Water Resources Division Washington 25, D.C. In reply refer to: September 20, 1990 GROUND WATER BRANCH MEMORANDUM NO. 90.08 SUBJECT: PUBLICATIONS--Release of report, "Preconditioned Conjugate-Gradient 2 (PCG2). A computer program for solving ground-water flow equations," by Mary C. Hill. The subject report has recently been published as Water-Resources Investigations Report 90-4048. The report describes and documents the program, PCG2, a numerical code which offers two previously unavailable preconditioned conjugate-gradient methods to solve the system of linear equations that are used to describe the groundwater flow system using the modular three-dimensional, finite-difference ground-water model (McDonald and Harbaugh, 1988, U.S. Geological Survey Techniques of Water Resources Investigations, Book 6, Chapter Al, A modular three-dimensional finite-difference ground-water flow model). The basic code of McDonald and Harbaugh (1988) gives the option of solving the system of linear equations using either the Strongly Implicit Procedure (SIP) or Slice-Successive Overrelaxation (SSOR). Logan Kuiper (1987, Computer program for solving groundwater flow equations by the preconditioned conjugate gradient method, U.S. Geological Survey Water-Resources Investigations Report 87-4091) added the option of using the method of preconditioned conjugate gradients and provided the choice of five types of preconditioners: three variants of incomplete Choleski factorization, the point Jacobi, and block Jacobi. Kuiper's report indicates which of the preconditoners tested were most successful in his experiments. One of the appeals to the conjugate gradient method is that it is not necessary to specify iteration parameters as is required for SIP and SSOR. The method of conjugate gradients is considered to be an N-step iterative method in that it is an algorithm applied to give successive approximations to a system of N linear equations that will yield the solution in at most N steps if computations could be done with complete accuracy. "Preconditioning" in association with the method of conjugate gradients, involves transforming the original system of linear equations into a system of linear equations that hopefully has the property that application of the conjugate gradient method to the transformed set of equations will result in fewer steps being needed to converge to a solution than would be needed for the original system of equations. Various preconditioners that may be used with the method of conjugate gradients have been identified in the "numerical" literature. The recently released computer code, PCG2, offers two preconditioners that previously have not been available for use with the subject modular finite-difference model (McDonald and Harbaugh, 1988): A modified incomplete Choleski factorization and a least-squares polynomial preconditioner. Mary Hill reports that compared to other preconditioners that require no greater computer storage, the results of her test results suggest that the modified incomplete Choleski factorization preconditioning is efficient on scalar computers; and the least squares polynomial preconditioner, while not as efficient as SIP or the modified incomplete Choleski on scalar computers, is more efficient on vector computers. The preconditioners that she considered were "those that produce a solver that has computer storage requirements less than or equal to the strongly implicit procedure (SIP) as programmed for the ground-water flow problem." It is worth noting that in numerical experiments reported on by Logan Kuiper (see "A comparison of iterative methods as applied to the solution of the nonlinear three-dimensional groundwater flow equation" by Logan K. Kuiper, SIAM J. Sci. Stat. Comput., vol 8, no. 4, p. 521-528, July 1987) the preconditioned conjugate gradient methods did better than SIP for most problems tested but he reported for the two-dimensional test problem (he considered 5 test problems) SIP did as well as the preconditioned conjugate- gradient methods; and Mary Hill (see "Solving ground water flow problems by conjugate-gradient methods and the strongly implicit procedure," by Mary Hill, Water Resources Research, vol. 26, no. 9, page 1961-1969, September 1990) found that SIP was sometimes more efficient than the preconditioned conjugate gradient methods that she tried for some three-dimensional and/or nonlinear flow problems. The results from these numerical studies suggest that although the preconditioned conjugate gradient methods have been shown to be efficient for many problems there are cases, that cannot be determined beforehand, for which SIP will be the better choice. In other words, no one method always works best. Water Resources Division employees may obtain the Computer Program, PCG2, from the Software Exchange System (SOFTEX) on the QVARSA Prime Computer. The SOFTEX entry identifier for PCG2 is COSALOOOO1. We recommend that this information be brought to the attention of technical and management personnel concerned with ground-water studies. (s) Thomas E. Reilly Acting Chief, Office of Ground Water WRD Distribution: A, B (memo only) S, FO, PO (memo and attachment)Release of report, "Preconditioned Conjugate-Gradient 2 (PCG2). A computer program for solving ground-water flow equations"