An upwinded state approximate Riemann solver

Abstract

Stability is achieved in most approximate Riemann solvers through ‘flux upwinding’, where the flux at the interface is arrived at by adding a dissipative term to the average of the left and right flux. Motivated by the existence of a collapsed interface state in the gas-kinetic Bhatnagar–Gross–Krook (BGK) method, an alternative approach to upwinding is attempted here; an interface state is arrived at by taking an upwinded average of left and right states, and then the flux is calculated as a function of this ‘collapsed’ interface state. This so called ‘state-upwinding’ approach gives rise to a new scheme called the linearized Riemann solver for the Euler and Navier–Stokes equations. The scheme is shown to be closely associated with the Roe scheme. It is, however, computationally less expensive and gives qualitatively comparable results over a wide range of problems. Most importantly, this scheme is found to preserve stationary contacts while not exhibiting the carbuncle phenomenon which plagues the Roe and other contact-preserving schemes. The scheme is therefore motivated as a new starting point to analyze the origin of the carbuncle phenomenon. Copyright © 2011 John Wiley & Sons, Ltd.