Accurate upwind schemes for the Euler equations

Abstract

A new upwind scheme that is nonoscillatory, uniformly second-order accurate, and computationally efficient is presented. The scheme consists of a reconstruction step followed by an upwind step. For the reconstruction step, a monotonicity constraint that preserves uniform second-order accuracy is introduced. The concept and coding of the constraint are simplified by the use of the median function. Computational efficiency is enhanced by devising a criterion that detects the smooth part of the data where the constraint is redundant. A slope-steepening technique, which has no effect a t smooth regions and can resolve a contact discontinuity in four cells, is shown. For the upwind step, Roe's flux-difference splitting is employed. I t is derived here as a flu-vector splitting, which has the advantage of conceptual simplicity. This derivation also leads to a straightforward entropy correction. Results of the new scheme are compared with those of the total variation diminishing (TVD) schemes, which are described below in a slightly simplified manner.