On the effect of temporal error in high-order simulations of unsteady flows

The present paper focuses on the effect that temporal error has on unsteady solutions modeled with a high-spatial-order, finite-element, streamline upwind/Petrov-Galerkin solver. Six L-stable, first- through fourth-order time-integration methods are discussed and exercised on two canonical test cases: the convecting isentropic vortex and the two-dimensional circular cylinder in crossflow. All results are shown on grids composed of P2 (third-order) elements. The isentropic vortex case is used to verify temporal order of accuracy, assess the relative split between dissipative and dispersive errors for each method with various time steps, and quantify the relative computational cost of each method. The results from the two-dimensional circular cylinder in crossflow simulations at various Reynolds numbers are compared to experimental results and accepted numerical results for vortex shedding frequency (Strouhal number), mean drag coefficient, and RMS lift and drag coefficient. The convergence of those parameters with respect to time step size is shown. These results highlight the benefits of high-order (greater than second order) time-integration methods and the need for time-step convergence studies on unsteady simulations.