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A Runge-Kutta-Newton-Krylov Algorithm for Fourth-Order Implicit Time Marching Applied to Unsteady Flows

Two implicit time-marching methods are investigated for accuracy and efficiency in solving the unsteady Navier-Stokes equations. The methods considered are the second-order backwards differencing formula and the fourth-order explicit-first-stage, single-diagonal-coefficient, diagonally-implicit Runge-Kutta method. First, the efficiency of two strategies for solving the nonlinear problem arising at each time step, an approximate factorization algorithm and a Newton-Krylov algorithm, is investigated. The Newton-Krylov strategy is seen to be more efficient, especially on fine meshes. Next, the relative efficiency of the two time-marching methods is compared for two-dimensional unsteady laminar flows over a cylinder and an airfoil. The backwards differencing method with approximate factorization dual time stepping is very efficient on a coarse mesh, whereas the implicit Runge-Kutta scheme combined with the Newton-Krylov algorithm is more efficient on finer meshes 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.