Accurate True Direction Solutions to the Euler Equations Using a Uniform Distribution Equilibrium Method

Abstract

A novel approach for the use of multiple continuous uniform distributions for reconstruction of the Maxwell-Boltzmann equilibrium probability distribution function is used for the solution of one and two dimensional Euler equations. The Uniform distribution Equilibrium Flux Method (UEFM) is a kinetic-theory based flux solver which calculates true directional, volume to volume fluxes based on integration (over velocity space and physical space) of a sum of uniform probability distribution functions working to approximate the equilibrium distribution function. The resulting flux expressions contain only the Heaviside unit step function and do not require the evaluation of the Exponential or Error Functions. The proposed method is verified using a series of one and two dimensional benchmarks and is shown to provide a higher level of accuracy (for a given computational expense) when compared to the similar Quiet Direct Simulation (QDS) method.