Convergence analysis of an implicit fractional-step method for the incompressible Navier–Stokes equations

In this paper, an implicit fractional-step method for numerical solutions of the incompressible Navier–Stokes equations is studied. The time advancement is decomposed into a sequence of two steps, and the first step can be seen as a linear elliptic problem; on the other hand, the second step has the structure of the Stokes problem. The two problems satisfy the full homogeneous Dirichlet boundary conditions on the velocity. At the same time, we introduce a diffusion term −θΔu in all steps of the schemes. It allows to calculate by the large time step and enhance numerical stability by choosing the proper parameter values of θ. The convergence analysis and error estimates for the intermediate velocities, the end-of step velocities and the pressure solution are derived. Finally, numerical experiments show that the feasibility and effectiveness of this method.