On convergence of implicit Runge-Kutta methods for the incompressible Navier-Stokes equations with unsteady inflow

This study investigates the convergence properties of implicit Runge-Kutta (IRK) methods when applied to the temporal solution of incompressible Navier-Stokes (N-S) equations with unsteady inflow. Owing to the differential-algebraic nature of spatially discretized N-S equations, conventional IRK methods may experience a significant order reduction while requiring exact satisfaction of the divergence-free constraint on the velocity field. Notably, the enhanced performance achieved through modified IRK techniques, such as projected Runge-Kutta methods and specialized Runge-Kutta methods, is confined to Runge-Kutta coefficients with specific attributes. In response to these limitations, this paper proposes a perturbed IRK scheme, modifying the intermediate stage equations of standard IRK methods by incorporating predefined perturbations, aiming to enhance convergence properties for the incompressible N-S equations accompanied by unsteady inflow. These perturbations within the scheme not only alleviate order reduction but also ensure exact enforcement of the divergence-free constraint. Moreover, the proposed scheme remains applicable in cases where the unsteady inflow is only available as discrete-time fields, rather than explicit functions of time. To demonstrate the efficiency of the proposed enhancement, an extensive analysis of the convergence properties for all considered IRK methods through a series of numerical experiments, is conducted.