The objective of this paper is to investigate the convergence properties of the Euler equation approximations. The theoretical investigation is carried out for linear hyperbolic equations in one and two dimensions. Using a Fourier decomposition, special attention is paid to the low and high frequencies. The spectra reveals that for stability and hence robustness of spatial discretizations with a small amount of numerical dissipation the grid transfer operators have to be accurate enough and the smoother of low temporal accuracy.
Numerical results give grid independent convergence in one dimension.
This is proved in special cases. For two-dimensional problems with a small amount of numerical dissipation, however, only a few grid levels contribute to an increased speed of convergence. This is explained by the small numerical dissipation leading to dispersion. Increasing the mesh density and hence making the problem over resolved increases the number of mesh levels contributing to an increased speed of convergence. If the steady state equations are elliptic, all grid levels contribute to the convergence regardless of the mesh density.