Implicit time integration schemes for the unsteady compressible Navier-Stokes equations: Laminar flow

The accuracy and efficiency of several lower and higher order time integration schemes are investigated for engineering solution of the discretized unsteady compressible Navier-Stokes equations. Fully implicit methods tested are either the backward differentiation formulas (BDF) or stage-order two. explicit, singly diagonally implicit Runge-Kutta (ESDIRK) methods. For this comparison an unsteady two-dimensional laminar flow problem is chosen: flow around a circular cylinder at Re = 1200. At temporal error tolerances consistent with engineering simulation. E approximate to 10(-1)-10(-2). first-order implicit Fuler (BDFI) is uncompetitive. While BDF3 is quite efficient, its lack of A-stability may be problematic in the presence of convection. At these same error tolerances, the fourth-order ESDIRK scheme is 2.5 times more efficient than BDF2. It is concluded that reliable integration is most efficiently provided by fourth-order Runge-Kutta methods for this problem where order reduction is not observed. Efficiency gains are more dramatic at smaller tolerances. (C) 2002 Elsevier Science (USA).