Robust Multigrid Methods for the Incompressible Navier-Stokes Equations
We introduce and compare several smoothers for the stationary incompressible Navier-Stokes equations in primitive variables on unstructured and locally refined grids. Special emphasis is laid on the robustness of the linear multigrid solver in view of lar
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Summary We introduce and compare several smoothers for the stationary incompressible Navier-Stokes equations in primitive variables on unstructured and locally refined grids. Special emphasis is laid on the robustness of the linear multigrid solver in view of large convecting velocities and bad aspect ratios in the grid. We describe a new streamwise numbering algorithm for the unknowns with a special treatment of the cyclic dependencies due to vortices. Further we describe the implemented smoothers and show diagrams of the convergence rates (per grid level) versus the aspect ratios of the elements. Additionally we present a comparison of some theoretical results for the Stokes-solution in a comer with our calculations.
1 Introduction and the Notion of "Robustness" The final goal of our recent work is a highly efficient solver for general CFD problems by using multigrid methods on locally refined and adapted unstructured meshes. The present paper is concerned with the first step on this way we have reached at We discuss the following problem: Since we use a linear multigrid as inner solver we want it to be able to yield good convergence rates for all problems the discretization passes to it Especially the cases of large convecting velocities (see section 2) and bad aspect ratios (see section 3) of the elements, which appear frequently in boundary layer fitted grids, should cause no severe problems for the smoother. To achieve this we tried the following strategy: Take some variant of ILU p as smoother and possibly decouple the equations by a transforming approach [WilJ. Then the algorithm should prove to be robust in view of bad aspect ratios. For the convection dominated case we will choose a special streamwise numbering of the unknowns.
2 Robustness in View of Large Convecting Velocities 2.1 Description of the numbering algorithm Let us switch off the diffusion for the time being. Due to the quasi Newton linearization and to the upwind scheme a given node depends only on its upwind neighbours. If we can find a global ordering of the unknowns in a way that the stiffness matrix has nonzero entries only in the lower triangle, then of course we will be able to solve the system of equations in one step even by a GauS-Seidel method. Unfortunately in most of the relevant cases there are vortices in the
216 W. Hackbusch et al. (eds.), Fast Solvers for Flow Problems © Springer Fachmedien Wiesbaden 1995
flow and therefore cyclic dependencies. But nevertheless we will obtain good results if we introduce arbitrary cuts through the vortices by removing just enough of the "cyclic" nodes to get rid of the cylic dependencies. We start the numbering at the inlet going in layers downstream but taking only nodes depending on the already numbered ones (those nodes will form the beginning of our new list). In a similar way we go upstream from the outlet (those nodes will make up the end of our new list). Finally we are left with nodes with cyclic dependencies. We cut it, appending those nodes to the beginning of our list Then ste
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