| Two time-marching schemes are investigated for accuracy and efficiency in solving the unsteady Navier-Stokes equations. The time-marching methods considered are the 2nd-order backwards differencing formula and the 4th-order explicit first stage, single diagonal coefficient, diagonally implicit Runge-Kutta method. The efficiency of approximate factorization and Newton-Krylov algorithms are investigated for solving the nonlinear problem arising at each iteration. Laminar two-dimensional flows around a cylinder and an airfoil are studied. The relative efficiency of the methods varies with the grid resolution. The backwards differencing method with approximate factorization dual time stepping is very efficient on the coarse grid, whereas the implicit Runge-Kutta scheme combined with the Newton-Krylov algorithm is the most efficient on the finer grids and when lower errors are required. The combination of the implicit Runge-Kutta method with the Newton-Krylov algorithm is shown to be very efficient for high-fidelity time-accurate simulations. |
