Comparison of Nested Factorization, Constrained Pressure Residual, and Incomplete Factorization Preconditionings
Authors: Behie A. 
Published: December 2007

Abstract
Two recently developed methods for the solution of the sparse block-banded linear equation sets generated by fully implicit reservoir simulators are investigated.
Nested factorization is a new approach to forming an incomplete factorization of the linear system. Comparisons are made of the nested factorization approach and the incomplete LU factorization (ILU) approach. Tests are done on both model problems and on problems generated by reservoir simulators. The nested factorization was no better than the best ILU method on both types of problems in most cases. In some cases it was considerably worse.
Constrained pressure residual preconditioning (CPR) is a variant of the COMBINATIVE method. These two methods are compared on problems generated by black oil and steam simulators. CPR gives small improvements in convergence rates in some cases.

Introduction
Accelerated incomplete LU factorization methods (ILU methods) have in recent years become the standard way to solve the block-banded, sparse linear equation sets encountered in reservoir simulation1,2,3. ILU methods combine an incomplete LU factorization of a fixed â??degreeâ?¿ and an acceleration method such as ORTHOMIN4 or conjugate gradient5 to provide an efficient robust iterative method, capable of handling problems which were traditionally thought to be too difficult to solve with an iterative solution method. Difficult problems may involve large permeability contrasts, large throughputs, a large number of grid blocks or large timestep sizes.
Nested factorization, a block incomplete factorization, has been described by Appleyard et al6,7. A nested factorization does not form strictly upper and lower factors but instead constructs block upper and lower factors. The nested factorization is compared in this paper to pointwise ILU factorization with different degrees of factorization and different orderings of the unknowns. The comparison is done both for the model problems discussed in references 6 and 7 and for problems generated by black oil and thermal simulators.
An extension of the ILU concept, the COMBINATIVE method, has been developed8 in which an estimate of the solution to only the pressure equations is found as well as an estimate of all unknowns via an LU factorization of the whole system. Both estimates are then followed by an acceleration method in a typical cycle of the process. A variant of this method known as constrained pressure residual preconditioning (CPR) has been developed by Wallis3. In this paper Wallis' vaiant is compared to the COMBINATIVE algorithm.

ILU Factorization
The accelerated incomplete factorization methods used in this report have been extensively described elsewhere1,2,9. They will be briefly summarized below.
An incomplete factorization LDU of a matrix A is defined by:Equation 1
where E is the error matrix and L, D and U are lower triangular, diagonal and upper triangular matrices respectively.
The number of off-diagonal bands in L and U is determined by the degree of the factorization1,2,9. A higher degree factorization produces a larger convergence rate but also requires more work per iteration to do the forward and backward solve. It also has a higher set-up cost and thus the method must converge faster to offset this.
The ordering of the grid nodes and in producing the incomplete factorization LDU affects the convergence rate1,2,9. Red-black ordering can be used to decouple half the unknowns leaving a system of half the original size to be solved iteratively.