A Spectral Study on the Dissipation and Dispersion of the WENO Schemes

The dissipation and dispersion (spectral) properties of the nonlinear fifth order classical weighted essentially non-oscillatory finite difference scheme (WENO-JS5) and its improved version (WENO-Z5) using the approximate dispersion relation (ADR) (Pirozzoli in J Comput Phys 219:489–497, 2006) and the nonlinear spectral analysis (NSA) (Fauconnier et al. in J Comput Phys 228(6):1830–1861, 2009) are studied. Unlike the previous studies, the influences of the sensitivity parameter in the definition of the WENO nonlinear weights are also included for completeness. The fifth order upwinded central linear scheme (UW5) serves as the reference and benchmark for the purpose of comparison. The spectral properties of the WENO differentiation operator is well predicted theoretically by the ADR and validated numerically by the simulations of the WENO schemes in solving the scalar linear advection equation. In a long time simulation with an initial broadband wave, the WENO schemes generate spurious high modes with amplitude and spread of wavenumbers depend on the value of the sensitivity parameter. The NSA is applied to investigate the statistical nonlinear behavior, due to the nonlinear stencils adaptation of the WENO schemes, with a large set of initial conditions consisting of synthetic scalar fields with a prescribed energy spectrum and random phases. The statistics indicate that there is a small probability of an existence of a mild anti-dissipation in the low wavenumber range regardless of the size of the sensitivity parameter. Numerical examples demonstrate that the WENO-Z5 scheme is not only less dissipative and dispersive but also less sensitive to random phases than the WENO-JS5 scheme. Furthermore, a sensitivity parameter adaptive technique, in which its value depends on the local smoothness of the solution at a given spatial location and time, is introduced for solving a linear advection problem with a discontinuous initial condition. The preliminary result shows that the solution computed by the sensitivity parameter adaptive WENO-Z5 scheme agrees well with those computed by the WENO-Z5 scheme and the UW5 scheme in regions containing discontinuities and smooth solutions, respectively.