An analysis of the upwind moment scheme and its extension to systems of nonlinear hyperbolic-relaxation equations

Abstract

The goal of this research is developing a unified numerical method for simulating continuum and transitional flow. To achieve our ultimate goal, first, hyperbolic-relaxation equations are introduced, then a new discretization method is developed. The method is based on Huynh’s upwind moment scheme, with implicit treatment of the source term. Our previous linear method is generalized to 1-D nonlinear hyperbolic-relaxation equations. First, a Fourier analysis is conducted to uncover the accuracy and stability. Then, the Euler equations with heat transfer, which reduce to the isothermal Euler equations in the equilibrium limit, are adopted as a model equation for the numerical experiment. The analysis and numerical results show the superiority of the proposed method in both accuracy and efficiency over the semi-discrete, method-of-line approach.