Convergence Acceleration for Multistage Time-Stepping Schemes

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 could be used. The implicit preconditioner addresses the stiness in the discrete equations associated with stretched meshes. Numerical dissipation operators (based on the Roe scheme, a matrix formulation, and the CUSP scheme) as well as the number of RK stages are considered in evaluating the RK/implicit scheme. Both the numerical and computational eciency of the scheme with the dieren t 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. In two dimensions, turbulent o ws over an airfoil at subsonic and transonic conditions are computed. The eects of mesh cell aspect ratio on convergence are investigated for Reynolds numbers between 5:7 10 6 and 100:0 10 6 . Results are also obtained for a transonic wing o w. For both 2-D and 3-D problems, it is demonstrated that the computational time of a well-tuned standard RK scheme can be reduced at least a factor of four.