The objective of the present work is to provide a theoretical analysis of the spectral properties of a selection of nonlinear finite difference schemes. The Nonlinear Spectral Analysis (NSA) of Fauconnier and Dick (2011)  is applied in order to study the statistical behavior of the modified wavenumber of TVD schemes in combination with five common limiter functions and also the 5th-order WENO scheme, for a large set of synthetic scalar fields with prescribed energy spectrum and random phase. The theoretical observations are verified numerically for the linear propagation of a broadband wave.
Fauconnier, D., & Dick, E. (2013). Spectral analysis of nonlinear finite difference discretizations. Journal of Computational and Applied Mathematics, 246, 113–121. https://doi.org/10.1016/j.cam.2012.11.004