Analysis of the spatial error for a class of finite difference methods for viscous incompressible flow

Several first- and second-order finite difference methods for incompressible flow based on prescribed forms of the discrete gradient and divergence operators are considered. Expansions for the spatial error for these methods are presented. So-called alternating expansions and numerical boundary layers are required to describe the errors arising from schemes with decoupled pressure approximations and regularizing terms, respectively. Alternating expansions in the discrete projection operator can be amplified by the viscous term and lead to a reduction in the accuracy of the computed pressure. These error expansions can be combined with simple stability estimates to show the convergence of the discrete solutions to the nonlinear time-dependent and steady problems when a discrete adjoint condition is satisfied. However, the analysis here does not consider the split-step nature of projection methods. Convergence order predictions are verified in a careful numerical study.