Extension of fractional step techniques for incompressible flows: The preconditioned Orthomin(1) for the pressure Schur complement

The objective of this paper is to present different fractional step schemes in the algebraic context to solve the incompressible Navier–Stokes equations, test them and pick the best one in terms of efficiency and robustness. The equivalence between fractional step schemes and iterative methods for the pressure Schur complement system has been well established in the literature. For example, the classical incremental projection scheme can be associated with a Richardson iteration for the pressure Schur complement plus a correction to enforce the mass conservation. We introduce in this paper an Orthomin(1) iteration which minimizes the Schur complement residual at each solver iteration by using, in the updating step, a factor dynamically computed. Two versions are considered, namely the momentum preserving and continuity preserving versions. The method is compared to the classical Richardson method, including the continuity and momentum preserving versions. In addition, two Schur complement preconditioners are considered and compared, based on the approximation of the weak Uzawa operator. From the implementation point of view, the benefit of the method is two fold. On the one hand, it can be easily implemented starting from the global matrix of the monolithic scheme, without changing the assembly. On the other hand, it enables the use of simple algebraic solvers without the need for complex preconditioners; this is a requirement for massively parallel computers. The four methods are finally tested and compared through the solution of numerical examples. The main conclusion is that with very few additional computation, the Orthomin(1) iteration largely improves the global convergence properties of the fractional schemes here presented.