Bram van Leer

Flux-Vector Splitting for the Euler Equation

When approximating a hyperbolic system of conservation laws w t + {f(w)} t = 0 with so-called upwind differences, we must, in the first place, establish which way the wind blows. More precisely, we must determine in which direction each of a variety of signals moves through the computational grid. For this purpose, a physical model of the interaction between computational cells is needed; at present two such models are in use.

Towards the ultimate conservative difference scheme III. Upstream-centered finite-difference schemes for ideal compressible flow

Abstract Finite-difference schemes for the conservation laws of ideal compressible flow are constructed on the basis of upstream-centered convective schemes, Fromms second-order scheme in particular. The upstream centering generates a number of higher-order terms, making the schemes quite complex. In consequence, they seem to compare unfavorably with central-difference schemes as regards computational efficiency. Previously derived upstream-centered terms that prevent numerical oscillations in Fromms scheme partly lose their effect when included in a version of the scheme for compressible flow. Apparently, the finite-difference approach is of little avail in formulating upstream schemes for compressible flow. It is anticipated that Godunovs approach, involving more of the physics in the conservation laws, will lead to more attractive schemes.

Characteristic time-stepping or local preconditioning of the Euler equations

A derivation is presented of a local preconditioning matrix for multidimensional Euler equations, that reduces the spread of the characteristic speeds to the lowest attainable value. Numerical experiments with this preconditioning matrix are applied to an explicit upwind discretization of the two-dimensional Euler equations, showing that this matrix significantly increases the rate of convergence to a steady solution. It is predicted that local preconditioning will also simplify convergence-acceleration boundary procedures such as the Karni (1991) procedure for the far field and the Mazaheri and Roe (1991) procedure for a solid wall.

Experiments with implicit upwind methods for the Euler equations

Abstract A number of implicit integration schemes for the one-dimensional Euler equations with conservative upwind spatial differencing are tested on a problem of steady discontinuous flow. Fastest convergence (quadratic for the first-order “backward Euler” scheme) is obtained with the upwind switching provided by van Leers differentiable split fluxes, which easily linearize in time. With Roes nondifferentiable split flux-differences the iterations may get trapped in a limit-cycle. This also happens in a second-order scheme with split fluxes, if the matrix coefficients arising in the implicit time-linearization are not properly centered in space. The use of second-order terms computed from split fluxes degrades the accuracy of the solution, especially if these are subjected to a limiter for the sake of monotonicity preservation. Second-order terms computed from the characteristic variables perform best.

Splitting of inviscid fluxes for real gases

Abstract Flux-vector and flux-difference splittings for the inviscid terms of the compressible flow equations are derived under the assumption of a general equation of state for a real gas in equilibrium. No unnecessary assumptions, approximations, or auxiliary quantities are introduced. The formulas derived include several particular cases known for ideal gases and readily apply to curvilinear coordinates. Applications of the formulas in a TVD algorithm to one-dimensional shock-tube and nozzle problems show their quality and robustness.

Upwind and High-Resolution Methods for Compressible Flow: From Donor Cell to Residual- Distribution Schemes †

In this paper I review three key topics in CFD that have kept researchers busy for half a century. First, the concept of upwind differencing, evident for 1-D linear advection. Second, its implementation for nonlinear systems in the form of high- resolution schemes, now regarded as classical. Third, its genuinely multidimensional implementation in the form of residual-distribution schemes, the most recent addition. This lecture focuses on historical developments; it is not intended as a technical review of methods, hence the lack of formulas and absence of figures.

Implicit flux-split schemes for the Euler equations

PROGRESS in the development of implicit algorithms for the Euler equations using the flux vector splitting method is described. Comparisons of the relative efficiency of relaxation and spatially split, approximately factored methods on a vector processor for transonic and supersonic twodimensional channel flows are made. A hybrid threedimensional algorithm is developed that uses relaxation in one coordinate direction and approximate factorization in the crossflow plane. The scheme is completely vectorizable and recovers conventional space-marching schemes for fully supersonic flows. The method is applied to a forebody shape in supersonic flow with an embedded pocket of subsonic flow.

A Discontinuous Galerkin Method for Diffusion Based on Recovery

We present the details of the recovery-based DG method for 2-D diffusion problems on unstructured grids. In the recovery approach the diffusive fluxes are based on a smooth, locally recovered solution that in the weak sense is indistinguishable from the discontinuous discrete solution. This eliminates the introduction of ad hoc penalty or coupling terms found in traditional DG methods. Crucial is the choice of the proper basis for recovery of the smooth solution on the union of two elements. Some results on accuracy, stability and the range of eigenvalues are given, together with numerical solutions on rectangular grids.

Inviscid flux-splitting algorithms for real gases with non-equilibrium chemistry

Several flux-splitting methods for the inviscid terms of the compressible-flow equations are derived for gases that are not in chemical equilibrium. Formulas are presented for the extension to chemical nonequilibrium of the Steger-Warming and Van Leer flux-vector splittings, and the Roe flux-difference splitting. The splittings are incorporated in a TVD algorithm and applied to one-dimensional shock-tube and nozzle flows of dissociating air, includung five species and 11 reaction steps for the chemistry.

Use of a rotated Riemann solver for the two-dimensional Euler equations

A scheme for the two-dimensional Euler equations that uses flow parameters to determine the direction for upwind -differencing is described. This approach respects the multi-dimensional nature of the equations and reduces the grid-dependence of conventional schemes. Several angles are tested as the dominant upwinding direction, including the local flow and velocity-magnitude-gradient angles. Roes approximate Riemann solver is used to calculate fluxes in the upwind direction, as well as for the flux components normal to the upwinding direction. The approach is first tested for two-dimensional scalar convection, where the scheme is shown to have accuracy comparable to a high-order MUSCL scheme. Solutions of the Euler equations are calculated for a variety of test cases. Substantial improvement in the resolution of shock and shear waves is realized. The approach is promising in that it uses flow solution features, rather than grid features, to determine the orientation for the solution method.