Numerical Error Estimation of Time Integration Schemes on Inviscid Flows around an Oscillating Airfoil

The actual time-accuracy of several time integration schemes is investigated for the unsteady compressible Euler equations. Runge-Kutta, implicit-approximate factorization, and implicit iterative schemes have been applied to several unsteady problems of transonic flows over a pitching airfoil. The flux difference splitting scheme of Roe with the MUSCL algorithm is used for determining numerical fluxes. The Runge-Kutta scheme produces little numerical error in its stable limit. However, no advantage of its third-order accurate form over its second-order accurate form can be found. The Beam-Warming scheme with approximate flux Jacobians is essentially first-order accurate in time and is not suitable for unsteady flow simulations. In order to preserve the accuracy of the numerical solutions, higher-order time-accurate forms as well as Newton-type iterations are necessary for the implicit time integration scheme.