On Some Numerical Dissipation Schemes

Several schemes for introducing an arti cial dissipation into a central di erence approximation to the Euler and Navier Stokes equations are considered. The focus of the paper is on the convective upwind and split pressure (CUSP) scheme, which is designed to support single interior point discrete shock waves. This scheme is analyzed and compared in detail with scalar dissipation and matrix dissipation (MATD) schemes. Resolution capability is determined by solving subsonic, transonic, and hypersonic ow problems. A nite-volume discretization and a multistage time-stepping scheme with multigrid are used to compute solutions to the ow equations. Numerical solutions are also compared with either theoretical solutions or experimental data. For transonic airfoil ows the best accuracy on coarse meshes for aerodynamic coe cients is obtained with a simple MATD scheme. The coarsegrid accuracy for the original CUSP scheme is improved by modifying the limiter function used with the scheme, giving comparable accuracy to that obtained with the MATD scheme. The modi cations reduce the background dissipation and provide control over the regions where the scheme can become rst order.