Multigrid for Steady Gas Dynamics Problems

This paper consists of two parts. In the rst part we give a review of a good multigrid method for solving the steady Euler equations of gas dynamics on a locally reened mesh. The method is self-adaptive and makes use of unstructured grids that can be considered as parts of a nested sequence of structured grids. It is brieey described and applied to some steady Euler-ow problems. The method appears to be much more accurate and ef-cient than the corresponding multigrid method that applies global reenements only. In the second part of the paper, vectorisation of the code is treated. To enable this vectorisa-tion, index arrays are introduced and added to the quad-tree type data-structure that is applied in the scalar case. Speed-up factors are given for the same test cases as considered in the rst part of the paper. The results are most satisfactory. 1 A solution-adaptive multi-grid method for the steady Euler equations In this rst section we describe a self-adaptive multigrid method, developed at CWI, for the solution of steady gas dynamics problems. The method can be applied for the Euler as well as for the Navier-Stokes equations. However, because of space limitations in this paper we restrict ourselves to the treatment of the Euler equations. The extension to the Navier-Stokes equations is found e.g. in 1]. For a survey of diierent multigrid approaches to these equations we refer to 2]. Our Euler solver essentially uses a sequence of nested reenements, and the discretisations at the various levels of reenement yield a set of nonlinear algebraic equations for each discretisation level. The method to solve these nonlinear systems is the nonlinear multigrid scheme (FAS), which may be embedded in the full multigrid algorithm (FMG) as well as in an iterative defect-correction process (ItDeC). First these three basic algorithms are described as they are combined with the use of locally reened grids. Next the strategy to introduce local reenements is described. Following that, aspects of the computer coding are discussed for the local grid-reenement method. As an illustration, numerical results are presented for shock reeection on a at surface and for transonic ow around an airfoil. These test cases are standard ; they are used to validate the method and to get an idea of the gain in eeciency by the local grid-reenement method, in comparison to the corresponding uniform-grid method. We give CPU execution times for the …