In computational fluid dynamics the Riemann problem, an initial value problem with piecewise constant initial data having only one discontinuity, plays a special role. An exact or an approximate solution to this problem, whose relevance for numerical computations was originally suggested in , is used in many modern codes for hyperbolic conservation laws to find the fluxes of conserved quantities, such as mass, momentum, and energy, across the interfaces of computational cells. Deriving the flux in such a way allows many properties of the governing equations to be directly built into the numerical method, in particular properties having to do with the directionality of wave propagation. For example, if all information propagates through an interface in only one direction, then one would want to evaluate the interface flux based only on the upstream state, so that the flux is wholly upwind. How can such a case be detected? One situation appears to be obvious. Suppose both initial states in a Riemann problem are supersonic in the same direction. Then, with all the information initially propagating in this direction, it seems natural to assume that it will continue to propagate in one direction even after the two states interact. In fact, this is exactly what many numerical schemes will do. In particular, it is done by any code that determines the domain of influence before solving the Riemann problem. Some writers even take this property to be a necessary condition for a flux function to be "properly upwind." In this note we want to point out that this notion of "proper upwinding" is erroneous.
From the theory of hyperbolic conservation laws it is well known (see, for example, [4]) that the speed of an entropy-satisfying "convex" wave lies in the range determined by the characteristics of an appropriate family immediately to the left and to the right of the wave. In the scalar case, which admits waves of only one family, this means that in a Riemann problem the characteristics computed from the initial data a priori determine the domain of influence to which the resulting wave will be confined. Thus, if the characteristics on both sides of the initial discontinuity are in the same direction, the interface flux can always be upwinded. This result is so natural that it is tempting to extend it to the non-scalar case as in Fig. . However, unless the initial states in a Riemann problem can be connected by a single wave, they are not neighboring states for t > 0. The possibility is therefore open that the domain of influence of the initial discontinuity extends outside the union of the domains of dependence defined by the two initial states (see Fig. ).
That this can indeed happen is easy to demonstrate for gas dynamics. Consider the following Riemann problem. Let the initial left state L be supersonic in the positive direction (see Fig. ). From the up diagram (see ) it is clear that for any such left state we can 611