Multigrid Methods for the Acceleration and the Adaptation of the Transonic Flow Problems

Abstract

In this paper the two of the three parts of a study towards the Block Adaptive Multigrid (BAM) method are presented. The BAM method is specially designed for implicit schemes with the aim to handle finite volume discretizations with the optimum accuracy at the minimum CPU-time and storage, points very critical when industrial problems are to solved. The first part is a thorough study of the multigrid schemes and their implementation into the single grid code, an implicit upwind code using a characteristic flux extrapolation scheme is exhibited. This results into a very efficient code capable to handle complex transonic viscous flow problems even by small workstations. As the correct implementation of the multigrid theory is essential whereas the adaptive method will rely on, whereas the grid independent convergence rate is the best proof for any inconsistencies.

With respect to the error analysis the prediction of the truncation error of the solution is considered. The calculation of the truncation error, although based theoretically on the multigrid procedure for the present approach, it can be used also without multigrid. Examples of the effectiveness of the prediction of the solution error are presented for common test cases, where a correlation between the predicted errors and the actual errors of the solution is established. Additionally, in order to prove the validity of the error prediction, crude adaptions with grid refinement procedures are used for regions with high truncation errors whereas the accuracy of the solution is improved considerably. The efficiency achieved by the combination of multigrid and the adaptive refinement procedure is extraordinary, but further studies on the adaptive strategies and the refinement grid solution are required. Moreover, the extension of the above methods for turbulent three dimensional problems is straight forward and very promising.