A STUDY of multigrid smoothers used in compressible CFD based on the convection diffusion equation

Abstract

We look at multigrid methods for unsteady viscous compressible flows. We specifically target smoothers that can be used in parallel and without computation of a Jacobian, which are particlarly attractive candidates in the context of Discontinuous Galerkin discretizations. In CFD, a plethora of nonlinear smoothers have been suggested which are hard to analyze. Our methodology is to use a linear model problem, here the convection diffusion equation, to be able to classify and compare smoothers better. Specifically, we consider explicit and implicit pseudo time iterations, GMRES as a smoother, SGS and implicit line smoothers. We relate GMRES to explicit Runge-Kutta smoothers, identify implicit line smoothers as Block Jacobi and analyze the potential of implicit pseudo time iterations. Finally, we discuss the relation between methods for steady and unsteady flows. Numerical results show that GMRES is a very attractive smoother in this context.