Temporal error analysis of a BDF2 time-discrete scheme for the incompressible Navier–Stokes equations with variable density

The paper is devoted to the study of the Navier–Stokes equations with variable density by reformulating the original equations. We investigate a BDF2 time-discrete scheme specifically designed for the numerical solution of variable density flows. A thorough unconditional stability analysis of this method is conducted, demonstrating its validity and robustness in handling such complex fluid dynamics problems. Additionally, we establish a rigorous proof of the second-order temporal convergence rate O(τ²) under certain regularity assumptions regarding the smoothness of the solutions, where τ represents the time step size. Finally, we present some numerical experiments to validate the analysis and demonstrate the effectiveness of this scheme.