FILU - an efficient and powerful preconditioner for reservoir simulation problems

Oleg Diyankov, Vladislav Pravilnikov, Serge Koshelev, and Natalia Kuznetsova

Preconditioned iterative methods provide some of the most efficient approaches for solving general, large sparse linear systems arising in many scientific and engineering applications. It is well-known that the convergence rate of an iterative solver depends strongly on the quality of the preconditioner used. Incomplete LU factorization is one of the most popular preconditioners for non-symmetric linear systems.

We present a FILU (Fill-in Incomplete LU) preconditioner with two drop tolerance parameters. In addition FILU has a fill-in threshold parameter and a strip tolerance. In combination with scalings and reorderings this preconditioner shows good results for problems arising in reservoir simulation used in the energy industry.

FILU implements a row-wise oriented ijk-variant of LU factorization. The first drop tolerance parameter is used for dropping only new elements which extend the original pattern.  The second drop tolerance parameter is used for dropping elements from a preconditioned row. The fill-in threshold parameter is used to limit the number of nonzero elements in a preconditioned row. The strip tolerance parameter is used for dropping "small" elements from the original pattern of a row before preconditioning.

Moreover, we have developed special norms different for the L and U parts of a preconditioned row. This extended set of dropping criteria with special row norms allows us to obtain a preconditioner with a good balance between quality of LU factorization and extension ratio of the original pattern. The performance of the preconditioner is illustrated by experiments with linear systems arising in hydrocarbon reservoir simulation.