A stabilized implicit fractional-step method for the time-dependent Navier–Stokes equations using equal-order pairs

A stabilized implicit fractional-step method for numerical solutions of the time-dependent Navier–Stokes equations is presented in this paper. The time advancement is decomposed into a sequence of two steps: the first step has the structure of the linear elliptic problem; the second step can be seen as the generalized Stokes problem. The two problems satisfy the full homogeneous Dirichlet boundary conditions on the velocity. On the other hand, a locally stabilized term is added in the second step of the schemes. It allows one to enhance the numerical stability and efficiency by using the equal-order pairs. Convergence analysis and error estimates for the velocity and pressure of the schemes are established via the energy method. Some numerical experiments are also used to demonstrate the efficiency of this new method.