Analysis of the anisotropy of group velocity error due to the application of spatial finite difference schemes to the solution of the 2D linear euler equations

Numerical differencing schemes are subject to dispersive and dissipative errors, which in one dimension are functions of wavenumber. When these schemes are applied in two or three dimensions, the errors become functions of both wavenumber and the direction of wave propagation. In this paper spectral analysis was used to analyse the magnitude and direction in error of the group velocity of vorticity-entropy and acoustic waves in the solution of the linearised Euler equations in a twodimensional Cartesian space. The anisotropy in these errors for three schemes were studied as a function of the wavenumber, wave direction, mean flow direction and mean flow Mach number. It was found that the traditional measure of error - the ratio of the magnitudes of the numerical to real group velocities - does not accurately capture the total error for waves which are traveling in an oblique direction to the mean flow. Therefore a second measure of a scheme's error that better represents the total error in the scheme is presented. Numerical experiments were run to provide confirmation of the developed theory.