Block factorized preconditioners for high‐order accurate in time approximation of the Navier‐Stokes equations

Computationally efficient solution methods for the unsteady Navier‐Stokes incompressible equations are mandatory in real applications of fluid dynamics. A typical strategy to reduce the computational cost is to split the original problem into subproblems involving the separate computation of velocity and pressure. The splitting can be carried out either at a differential level, like in the Chorin‐Temam scheme, or in an algebraic fashion, like in the algebraic reinterpretation of the Chorin‐Temam method, or in the Yosida scheme (see 1 and 19). These fractional step schemes indeed provide effective methods of solution when dealing with first order accurate time discretizations. Their extension to high order time discretization schemes is not trivial. To this end, in the present work we focus our attention on the adoption of inexact algebraic factorizations as preconditioners of the original problem. We investigate their properties and show that some particular choices of the approximate factorization lead to very effective schemes. In particular, we prove that performing a small number of preconditioned iterations is enough to obtain a time accurate solution, irrespective of the dimension of the system at hand. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 487–510, 2003