American Institute of Aeronautics and Astronautics 1 Application of a line-implicit scheme on stretched unstructured grids Peter Eliasson 1 FOI, Swedish Defence Research Agency, SE-16490 Stockholm, Sweden Per Weinerfelt 2 Saab Aerosystems, SE-581 88 Linköping, Sweden and Jan Nordström 3 FOI, Swedish Defence Research Agency, SE-16490 Stockholm, Sweden Uppsala University, Department of Information Technology, SE-751 05 Uppsala, Sweden A line-implicit Runge-Kutta time stepping scheme is derived, implemented and applied. It is applied to fluid flow problems governed by the Navier-Stokes equations on stretched unstructured grids. The flow equations are integrated implicitly in time along structured lines in regions where the grid is stretched, typically in the boundary layer, and explicitly elsewhere. The integration technique is introduced for steady state problems with the intention to speed up the rate of convergence. It is extended to unsteady problems by a dual time stepping approach. The paper focuses on the implementation of the line-implicit scheme starting from an explicit multigrid flow solver and on the application of it. Numerical results are presented for test cases in two and three dimensions for inviscid and viscous flow problems. The line-implicit time integration convergence rates are compared to pure explicit convergence rates and the gain is quantified in terms of reduction of iterations and CPU time. All presented test cases show improved convergence rates. The gain is highest for the three dimensional test cases for which reductions of up to 75% of the computing time is obtained. I. Introduction OMPUTATIONAL fluid dynamics problems in general have widely varying length and time scales. In three dimensional viscous problems with several millions of grid points, the length scales spanning the computational domain are several orders of magnitude larger than the smallest scales resolved by neighboring points, for example in the boundary layer. To resolve these scales, anisotropic grids are used with high aspect ratio cells in boundary layers and possibly in wakes which results in stream-wise length scales that are several orders of magnitude larger than the normal length scales. It is well known that explicit methods applied to problems with disparate scales may be extremely inefficient due to the restrictions in the time step coming from the smallest scale. Implicit methods, on the other hand, can handle these problems efficiently since they have no stability restriction on the time step. The only significant problem with the implicit technique is that a very efficient solver is required. Explicit Runge-Kutta methods in combination with multigrid have shown to be efficient for inviscid fluid dynamics problems 1-3 where the computational grids have moderate stretching and steady state convergence may be obtained in O(10 2 ) iterations. For viscous high Reynolds number flows, stretched and highly anisotropic grids are used and the convergence is degraded due to the small scales introduced by the stretching. The number of iterations 1 Deputy Research Director, Department of Computational Physics, AIAA Member. 2 PhD., Aeronautical Engineering. 3 Director of Research in Numerical Analysis at FOI and Adjunct Professor in Numerical Analysis at Uppsala University. C