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, ϵ≈10−1–10−2, first-order implicit Euler (BDF1) 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.