A CFD Euler solver from a physical acoustics–convection flux Jacobian decomposition

Abstract

We introduce 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