Robustness and Efficiency of an Implicit Time-Adaptive Discontinuous Galerkin Solver for Unsteady Flows

High-fidelity fluid dynamics simulations of unsteady flows are nowadays of great interest for many industrial fields. This class of simulations, as they are characterized by a wide range of temporal scales, requires robust, accurate and efficient long time integration strategies. These features can be achieved by an appropriate coupling of high-order time integration schemes and time-step adaptation algorithms. The adaptation algorithms are typically based on a local error estimator, which exploits the local truncation error of the time integration scheme and of its lower order embedded scheme. In literature few information are available to assess the benefits in terms of robustness, accuracy, and efficiency provided by the coupling between temporal schemes and adaptation strategies for unsteady CFD simulations. The aim of this work is to reduce this gap, presenting a numerical investigation of the performance for different adaptive time-step strategies, based on implicit Rosenbrock-type temporal schemes, in a high-order discontinuous Galerkin solver. The performance of the considered time integration strategies for the autonomous ODE system resulting from the DG space discretization of the Navier–Stokes equations is assessed for several test cases of increasing stiffness and difficulty, identifying the best scheme and algorithm: ( i ) the 2D laminar flow around a circular cylinder and around a tandem of cylinders at Re D = 100 ; ( ii ) the 2D viscous flow through a porous media, modelled as an array of cylinders, at Re D = 2,100 and Re D = 10,000 ; ( iii ) the 3D turbulent flow through a 4-wheels rudimentary landing gear (RLG) at Re D = 1 ×10^6.