Upwind schemes for the Navier-Stokes equations

A discretization method is presented for the full, steady, compressible Navier-Stokes equations. The method makes use of quadrilateral finite volumes and consists of an upwind discretization of the convective part and a central discnetization of the diffusive part. In the present paper, the emphasis lies on the discretization of the convective part. The applied solution method directly solves the steady equations by means of a Newton method, which requires the discretization to be continuously differentiable. For two upwind schemes which satisfy this requirement (Osher's and van Leer's scheme}, results of a quantitative error analysis are presented. Osher's scheme appaars to be more and more accurate than van Leer's scheme with increasing Reynolds number. A suitable higher-order accurate discretization of convection is chosen. Based on this higher-order scheme, a new limner is constructed. Further, for van Leer's scheme, a solid wall - boundary condition treatment is proposed, which ensures a continuous transition from the Navier-Stokes flow regime to the Euler flow regime. Numerical results are presented for a subsonic flat plate flow and a supersonic flat plate flow with oblique shock wave - boundary layer interaction. The resu Its obtained agree with the predictions made. Useful properties of the discretization method are that it allows an easy check of false diffusion and that it needs no tuning of parameters

Upwind discretization of the steady Navier-Stokes equations

A discretization method is presented for the full, steady, compressible Navier-Stokes equations. The method makes use of quadrilateral finite volumes and consists of an upwind discretization of the convective part and a central discretization of the diffusive part. In the present paper the emphasis lies on the discretization of the convective part. The solution method applied solves the steady equations directly by means of a non-linear relaxation method accelerated by multigrid. The solution method requires the discretization to be continuously differentiable. For two upwind schemes which satisfy this requirement, results of a quantitative error analysis are presented. Osher's scheme appears to be increasingly more accurate than van Leer's scheme with increasing Reynolds number. A suitable higher-order accurate discretization of the convection terms is derived. On the basis of this higher-order scheme, to preserve monotonicity, a new limiter is constructed. Additional aspects of the subject are discussed

Error analysis for function representation by the sparse-grid combination technique

Detailed error analyses are given for sparse-grid function representations through the combination technique. Two- and three-dimensional, and smooth and discontinuous functions are considered, as well as piecewise-constant and piecewise-linear interpolation techniques. Where appropriate, the results of the analyses are verified in numerical experiments. Instead of the common vertex-based function representation, cell-centered function representation is considered. Explicit, pointwise error expressions for the representation error are given, rather than order estimates. The paper contributes to the theory of sparse-grid techniques

A level-set method for moving material-void interfaces

This report is a feasibility study of a level-set method for the computation of moving interfaces, in an Eulerian formulation. The report briefly introduces level-set methods and focuses on the development of such a method for moving material-void interfaces. Results are presented for illustrative model problems. As concerns its ability to improve the geometrical resolution of free boundaries, the level-set method appears to perform excellently. Concerning the improvement of other than merely geometrical free-boundary properties, the method performs very well for downstream-facing fronts and is promising for upstream-facing ones