Analytic evaluation of finite difference methods for compressible direct and large eddy simulations

We investigate the performance of several different numerical methods using the von Neumann method for the compressible Euler Equations. The results are used to analytically estimate the error due to the numerical schemes with an approximate spectrum of the turbulence obtained from isotropic homogeneous turbulence theory. The final spectrum is calculated from the linearized von Neumann analysis and a characteristic time scale of turbulence. In this study, we compare the MacCormack scheme. (second and fourth order), the original Flux-Corrected-Transport (FCT) scheme, a second-order FCT scheme, a fourth-order Runge-Kutta, centered-space scheme, a second order implicit upwind scheme, a high order implicit upwind method and a recent sixth-order Pade method. Most methods have been used in prior studies involving large eddy or direct numerical simulations. The results show significant differences in the stability and resolution characteristics of the various methods. In conjunction with conventional tests for compressible flow methods (such as performance of the Riemann problem), the present approach may provide improved a priori evaluation of proposed methods for LES and DNS investigations.