Exact Nonlinear Spectral Analysis of Nonsmooth WENO-JS Solutions

Abstract

The fifth-order accurate Weighted Essentially Nonoscillatory space discretization developed by Jiang and Shu (WENO-JS) is studied theoretically. An exact Nonlinear Spectral Method (NSM) is developed based on an innovative yet simple methodology. The NSM explains the behaviour of nonsmooth solutions because it is valid for arbitrary modified wave numbers (MWN). The NSM clarifies the effects of the time integration methods and the Courant number. The mode isolation assumption, extensively used to analyse WENO-JS, is elucidated, and several novel findings are presented. The improved performance of the combination of WENO-JS with the forward Euler time integration method, compared to the Linear Fifth-Order Upwind discretization, is illustrated. The overdamping of the combination of WENO-JS with the popular third-order total variation diminishing Runge-Kutta method is discovered. Thus, the NSM covers several gaps in the current literature.