Investigation and Comparison of Implicit Smoothers Applied in Agglomeration Multigrid

Abstract

A straightforward implicit smoothing method implemented in several codes solving the Reynolds-averaged Navier–Stokes equations is the lower–upper symmetric Gauss–Seidel method. It was proposed several years ago and is attractive to implement because of its low memory requirements and low operation count. Since, for many examples, often a severe restriction of the Courant–Friedrichs–Lewy number and loss of robustness are observed, it is the goal of this paper to revisit the lower–upper symmetric Gauss–Seidel implementation and to discuss several alternative implicit smoothing strategies used within an agglomeration multigrid for unstructured meshes. The starting point is a full implicit multistage Runge–Kutta method. Based on this method, several additional features and simplifications are developed and suggested, such that the implicit method is applicable to high-Reynolds-number viscous flows; that is, the required matrices fit into the fast memory of the cluster hardware and the arising linear systems can be approximately solved efficiently. To this end, the focus is on simplifications of the Jacobian, as well as efficient iterative approximate solution methods. To significantly improve the approximate linear solution methods, grid anisotropy is taken care of for approximately solving both the linear systems and the agglomeration strategy. The procedure creating coarse-grid meshes is extended by strategies identifying structured parts of the mesh. This seems to improve the quality of coarse-grid meshes in the way that an overall better reliability of the multigrid can be observed. Furthermore, grid information is exploited within the iterative solution methods for the linear systems. Numerical examples demonstrate the gain with respect to reliability and efficiency.