Upwind finite element schemes based on the Petrov-Galerkin method; SUPG by Brooks and Hughes, optimal upwind method by Chee and Kwon and exponential weighting function method by Kakuda and Tosaka, are compared in this paper to investigate property of schemes and performances. The optimal upwind method is applied to the transonic flow and the exponential weighting function method is newly extended to the compressible flow region in this paper. The implemented schemes have the following : Petrov-Galerkin formulation with an oscillation-free shock-capturing operator, structured grid and explicit time integration with local-time-stepping.
These schemes are tested and compared on subsonic, transonic and supersonic compressible flow problems. Further, the effect of the control parameters on the accuracy is studied in some details and it reveals that the quality of solution is dependent on the performance of the shock capturing of each method. For good solutions with the SUPG method, the control parameters may be set to be 1 in the subsonic and transonic flow regions, but less than 1 may be required in the supersonic flow region. With the optimal upwind method, however, the control parameter for the full flow region is not needed. It is shown that the exponential weighting method does not give reasonable solutions in the subsonic flow region, but gives good solutions in the transonic and supersonic flow regions. In the transonic flow region, the SUPG method captures shock within three points, while the optimal upwind and the exponential weighting function methods capture shock within only two points.
