Convergence acceleration of Runge–Kutta schemes for solving the Navier–Stokes equations

The convergence of a Runge–Kutta (RK) scheme with multigrid is accelerated by preconditioning with a fully implicit operator. With the extended stability of the Runge–Kutta scheme, CFL numbers as high as 1000 can be used. The implicit preconditioner addresses the stiffness in the discrete equations associated with stretched meshes. This RK/implicit scheme is used as a smoother for multigrid. Fourier analysis is applied to determine damping properties. Numerical dissipation operators based on the Roe scheme, a matrix dissipation, and the CUSP scheme are considered in evaluating the RK/implicit scheme. In addition, the effect of the number of RK stages is examined. Both the numerical and computational efficiency of the scheme with the different dissipation operators are discussed. The RK/implicit scheme is used to solve the two-dimensional (2-D) and three-dimensional (3-D) compressible, Reynolds-averaged Navier–Stokes equations. Turbulent flows over an airfoil and wing at subsonic and transonic conditions are computed. The effects of the cell aspect ratio on convergence are investigated for Reynolds numbers between 5.7 × 10 6 and 100 × 10 6 . It is demonstrated that the implicit preconditioner can reduce the computational time of a well-tuned standard RK scheme by a factor between 4 and 10.