Stability of a Crank-Nicolson Pressure Correction Scheme Based on Staggered Discretizations

International audienceIn the context of Large Eddy Simulation of turbulent flows, the control of kinetic energy seems to be an essential requirement for the numerical scheme. Designing such an algorithm, ie as less dissipative as possible while being simple, for the resolution of variable density Navier-Stokes equations is the aim of the present work.The developed numerical scheme, based on a pressure correction technique, uses a Crank-Nicolson time discretization and a staggered space discretization relying on the Rannacher-Turek finite element. For the inertia term in the momentum balance equation, we propose a finite volume discretization, for which we derive a discrete analogue of the continuous kinetic energy local conservation identity. Contrary to what was obtained for the backward Euler discretization, the dissipation defect term associated to the Crank-Nicolson scheme is second order in time. This behaviour is evidenced by numerical simulations