Accurate Upwind Methods for the Euler Equations

Abstract

A new class of piecewise linear methods for the numerical solution of the one-dimensional Euler equations of gas dynamics is presented. These methods are uniformly second-order accurate and can be considered as extensions of Godunov’s scheme. With an appropriate definition of monotonicity preservation for the case of linear convection, it can be shown that they preserve monotonicity. Similar to Van Leer’s scheme, they consist of two key steps: a reconstruction step followed by an upwind step. For the reconstruction step, a monotonicity constraint that preserves uniform second-order accuracy is introduced. Computational efficiency is enhanced by devising a criterion that detects the “smooth” part of the data where the constraint is redundant. The concept and coding of the constraint are simplified by the use of the median function. A slope-steepening technique, which has no effect at smooth regions and can resolve a contact discontinuity in four cells, is described. As for the upwind step, existing and new methods are applied in a manner slightly different from those in the literature. These methods are derived by approximating the Euler equations via linearization and diagonalization. At a “smooth” interface, for economy, the linearization employs the arithmetic average of the left and right states, and the flux is obtained by one of the four simple models: the primitive-variable splitting, a simplified flux splitting, the one-intermediate-state model, or the central difference with artificial viscosity. Near a discontinuity, Roe’s flux-difference splitting is used. The current presentation of Roe’s method, via the conceptually simple flux-vector splitting, not only establishes a connection between the two splittings, but also leads to an admissibility correction with no conditional statement and an efficient approximation to Osher’s approximate Riemann solver. These reconstruction and upwind steps result in schemes that are uniformly second-order accurate and economical at smooth regions and yield high resolution at discontinuities.