Aytekin Çıbık

Aytekin Çıbık, Medine Demir, Songul Kaya

This study presents an efficient, accurate, effective and unconditionally stable time stepping scheme for the Darcy-Brinkman equations in double-diffusive convection. The stabilization within the proposed method uses the idea of stabilizing the curvature for velocity, temperature and concentration equations. Accuracy in time is proven and the convergence results for the fully discrete solutions of problem variables are given. Several numerical examples including a convergence study are provided that support the derived theoretical results and demonstrate the efficiency and the accuracy of the method.

Aytekin Çıbık, Fatma G. Eroglu, Songul Kaya

This paper investigates the long time stability behavior of multiphysics flow problems, namely the Navier-Stokes equations, natural convection and double-diffusive convection equations with an extrapolated blended BDF time-stepping scheme. This scheme combines the two-step BDF and three-step BDF time stepping schemes. We prove unconditional long time stability theorems for each of flow systems. Various numerical tests are given for large time step sizes in long time intervals in order to support theoretical results.

Aytekin Çıbık, Rui Fang

Continuous data assimilation (CDA) nudges observational data into governing equations to recover the underlying flow and improve predictions. Existing rigorous CDA analyses focus primarily on incompressible flows, yet no physical flow is perfectly incompressible. Approximating a slightly compressible flow with an incompressible model introduces non-negligible model errors. Data assimilation for compressible flows remains challenging due to strong nonlinearities and the presence of shocks. We design an algorithm that addresses the limitations of velocity-only nudging for slightly compressible flow. This work incorporates both velocity and pressure data from the slightly compressible flow and nudges both quantities into the incompressible Navier--Stokes equations. Our analysis shows that the model error decays exponentially in the initial error, with an asymptotic residual of order $\mathcal{O}(H)$, where H denotes the observation resolution. The analysis also identifies a scaling for the pressure nudging parameter $μ_1 = O(1/H^2)$ that ensures effective assimilation. We validate the theoretical results through a suite of numerical experiments: a convergence study confirming optimal rates, a modified Taylor--Green vortex benchmark demonstrating synchronization of energy, enstrophy, and pressure, and an acoustic wave propagation test that isolates the role of pressure nudging and achieves a $97.9\%$ reduction in pressure error relative to velocity-only assimilation. Together, these results provide a foundation for discrete error estimates and realistic compressible applications.