A structure-preserving projection method with formal second-order accuracy for the incompressible Navier–Stokes equations

The incompressible Navier–Stokes equations play an important role in describing extensive fluid phenomena in science and engineering. With some specific boundary treatments, the Navier–Stokes equations can satisfy an energy evolutional structure with respect to kinetic energy and works done by external forces. If the external forces are absent, the energy dissipation law is obtained. This work aims to develop a two-stage projection method which not only preserves the energy dissipation property (or energy stability) but also has a formal second-order accuracy in time. In the first stage, we calculate the auxiliary velocity field and pressure with a first-order semi-implicit projection scheme and a half time step. In the second stage, the final velocity field and pressure are updated with a linear semi-implicit Crank–Nicolson-type projection method, where the auxiliary velocities are used to explicitly treat the advection term. To satisfy the energy law, we introduce an extra time-dependent supplementary variable in our proposed scheme. The evolutions of time-dependent supplementary variable are explicit and totally decoupled with velocities and pressure. During each time increment, it is only necessary to solve several linear elliptic type equations for velocities and two Poisson equations for pressure. Therefore, the numerical algorithm can be easily implemented. Moreover, we analytically prove the time-discretized energy dissipation law with respect to a total energy. The computational simulations indicate that the proposed method has desired accuracy and stability. The capabilities are further validated via the tests of lid-driven cavity flow, rotating vortex pairs, and flows passing through solid obstacles.