Time accurate simulations of supersonic unsteady flow

In this paper we study the properties of an implicit time integration method for the simulation of unsteady shock boundary layer interaction dow. Using an explicit second order Runge-Kutta scheme we determine a reference solution for the implicit second order Crank Nicolson scheme. This A-stable scheme allows the time step to be determined solely by the resolution requirements of the unsteady how phenomena. The nonlinear equations which result from the temporal and spatial discretisation are solved iteratively by adding a pseudo time derivative to which the Euler backward scheme is applied. As a criterion for the accuracy of the solution we relate the global error caused by the temporal integration to the error resulting from the spatial discretisation. We study the dependence of the accuracy of the solution on the time step and the accuracy with which the solution is determined at each instant. Numerical simulations show that the time step needed for acceptable accuracy is considerably larger than the explicit stability time step. For mean flow quantities the time step can be increased by a factor eighty while instantaneous flow quantities are predicted accurately with a twenty times larger step size.