| Abstract: | This paper introduces a continuum, i.e. non‐discrete, upstream‐bias formulation that rests on the physics and mathematics of acoustics and convection. The formulation induces the upstream‐bias at the differential equation level, within a characteristics‐bias system associated with the Euler equations with general equilibrium equations of state. For low subsonic Mach numbers, this formulation returns a consistent upstream‐bias approximation for the non‐linear acoustics equations. For supersonic Mach numbers, the formulation smoothly becomes an upstream‐bias approximation of the entire Euler flux. With the objective of minimizing induced artificial diffusion, the formulation non‐linearly induces upstream‐bias, essentially locally, in regions of solution discontinuities, whereas it decreases the upstream‐bias in regions of solution smoothness. The discrete equations originate from a finite element discretization of the characteristic‐bias system and are integrated in time within a compact block tridiagonal matrix statement by way of an implicit non‐linearly stable Runge–Kutta algorithm for stiff systems. As documented by several computational results that reflect available exact solutions, the acoustics–convection solver induces low artificial diffusion and generates essentially non‐oscillatory solutions that automatically preserve a constant enthalpy, as well as smoothness of both enthalpy and mass flux across normal shocks. Copyright © 1999 John Wiley & Sons, Ltd. |
