Efficient matrix decomposition for implicit algorithms

Abstract

Implicit procedures for solving the equations of computational fluid dynamics have been developed and used widely during the past quarter century. They all attempt to solve a matrix equation formed by partial linearization and discretization of the governing fluid flow equations. The practical ones in use today for multidimensional problems first approximate the original matrix equation with a "pre-conditioned" matrix equation that can be solved efficiently. Two such procedures are SIP, or strongly Implicit Procedure for L-U decomposition, originally presented by Stone for solving the heat equation in two and three dimensions, and AF, or spacial Approximate Factorization, used centrally in the Beam-Warming and the Brily-McDonald methods. They each have distinct advantages and disadvantages. A small modification to either, to be presented herein, can produce a new factorization procedure containing the advantages of each and the disadvantages of neither.