Application of a Line-Implicit Scheme on Stretched Unstructured Grids

Abstract

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.