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Compressible Navier-Stokes equations, which govern unsteady flow phenomena, have been numerically integrated with several high order ESDIRK schemes (Explicit first stage, Stiffly accurate Diagonally Implicit Runge-Kutta). This type of numerical scheme allows for embedded error estimation and, based on this error estimation, for adaptive time stepping, cf. [1, 2, 3]. In this paper different controller types have been applied for the automatic adaption of the time step size to a prescribed accuracy level: I-, PI-, PCand PID-controller. Depending on the choice of corresponding controller parameters, the simulation time can be varied considerably. On the one hand, a conservative choice leads to slow time step adaption, small step size and long simulation time. On the other hand, an aggressive parameter setting can lead to quick step size adaption, but also to rejected computations, due to large time step size, which does not keep the prescribed accuracy. This also may result in long simulation time. Obviously, there must be appropriate combinations of controller parameters, which minimize the simulation duration. Hence, a systematic study on a large controller parameter space has been conducted and regions with appropriate combinations have been identified. Both generic and complex test cases have been investigated. Overall CPU times and number of time steps are compared among different ESDIRK schemes and controllers. For the generic test case, most schemes show a narrow space for the choice of suitable controller parameters, but only one scheme shows good simulation performance without large sensitivities to parameter combinations. For the complex test cases, all schemes and controllers are less sensitive to parameter combinations. Furthermore, the optimum combination varies strongly with the type of governing equations.

APPROPRIATE COMBINATIONS OF CONTROLLER PARAMETERS FOR UNSTEADY FLOW SIMULATIONS WITH ADAPTIVE TIME STEP CONTROL