Insights from von Neumann analysis of high-order flux reconstruction schemes

The flux reconstruction (FR) approach unifies various high-order schemes, including collocation based nodal discontinuous Galerkin methods, and all spectral difference methods (at least for a linear flux function), within a single framework. Recently, an infinite number of linearly stable FR schemes were identified, henceforth referred to as Vincent–Castonguay–Jameson–Huynh (VCJH) schemes. Identification of VCJH schemes offers significant insight into why certain FR schemes are stable (whereas others are not), and provides a simple prescription for implementing an infinite range of linearly stable high-order methods. However, various properties of VCJH schemes have yet to be analyzed in detail. In the present study one-dimensional (1D) von Neumann analysis is employed to elucidate how various important properties vary across the full range of VCJH schemes. In particular, dispersion and dissipation properties are studied, as are the magnitudes of explicit time-step limits (based on stability considerations). 1D linear numerical experiments are undertaken in order to verify results of the 1D von Neumann analysis. Additionally, two-dimensional non-linear numerical experiments are undertaken in order to assess whether results of the 1D von Neumann analysis (which is inherently linear) extend to real world problems of practical interest.