In this paper, the issue of postshock oscillations generated by shock-capturing schemes is investigated. Although these oscilla-sional grid (Fig. 1 (left)). A first-order finite-volume tions are frequently small enough to be ignored, there are contexts scheme with a flux function corresponding either to the such as shock-noise interaction where they might prove very intruexact solution of a Riemann problem [7] or else to Roe's sive. Numerical experiments on simple nonlinear 2 ฯซ 2 systems of [30] approximate solution does achieve this resolution for conservation laws are found to refute some earlier conjectures on a steady one-dimensional shock. Recently, Jameson [12] their behavior. The trajectories in phase space of a computed state passing through a captured shock suggest the underlying mecha-and Liou [18] have called attention to economical classes nism that creates these oscillations. The results reveal a flaw in of flux functions that share this property. The Osher flux the way that the concept of monotonicity is extended from scalar function [25] gives rise to a family of shocks with two conservation laws to systems; schemes satisfying this formal condi-(exceptionally one) interior values (Fig. 1 (right)), as does tion fail to prevent oscillations from being generated, even for the flux-vector splitting scheme of van Leer [36]. These monotone initial data. This indicates that satisfactory design criteria do not exist at the present time that would ensure captured shocks properties are inherited by most modern high-order that are both narrow and free from oscillations. แฎ 1997 Academic Press schemes using the same flux functions, because such schemes usually revert to their first-order counterparts sufficiently close to the shocks. 1 1 It is possible to prove [6] that a flux function F(u L , u R ) which gives give a nice interpretation of the von Neumann method rise to one-point shocks cannot be a differentiable function of its input and its relation to modern techniques.
states. Since differentiability is a requirement for certain convergence techniques like Newton's method, the pursuit of maximal resolution may exact a toll. In practice, the question of convergence may be so compli-* Current address: Morgan Stanley & Co. Inc., 750 Seventh Ave., 7th floor, New York, NY 10019.
cated by other issues that this may not always be an important issue.