HIGH-ORDER TIME-ACCURACY USING APPROXIMATE JACOBIANS FOR THE COMPRESSIBLE NAVIER-STOKES EQUATIONS

It is well established that the most efficient implicit methods for solving compressible Navier-Stokes problems to steady state are approximations of Newton methods, using simplified Jacobians and inexact solution of the linear system at each iteration. Two such methods are described, which are cheap and stable for large CFL numbers. However, because of the approximate nature of the implicit operators they can not normally be applied directly to time-accurate calculations, but must be wrapped within a dual-time iteration, at the cost of efficiency. Here a time-accurate Runge-Kutta (RK) method is presented which overcomes this limitation. The explicit RK method is constructed in such a way that an arbitrary implicit term may be added at each stage without loss of order of accuracy in time. This term may then be chosen to stabilize the iteration at low cost- for which the previously mentioned approximate Newton operators are appropriate. The scheme is a special case of a Rosenbrock W-method. Significant performance gains are demonstrated over dual-time, and the need for time-step control is identified. 1