Among the insights shared: "The ideal numerical method for LES should have very low dissipation AND dispersion errors over the resolvable range of wave numbers, but dissipative for non-resolvable high wave numbers. In this way, the simulation will resolve a wide turbulent spectrum, while damping out the non-resolvable small eddies to prevent energy pile-up, which can drive the simulation divergent."

We recently analysed and compared several 6th-order spatial schemes for LES: the standard central FD, the upwind-biased FD, the filtered compact difference (FCD), and the discontinuous Galerkin (DG) schemes, with the same time integration approach (an Runge-Kutta scheme) and the same time step. The FCD schemes have an 8th order filter with two different filtering coefficients, 0.49 (weak) and 0.40 (strong). We first show the results for the linear wave equation with 36 degrees-of-freedom (DOFs) in Figure 1. The initial condition is a Gaussian-profile and a periodic boundary condition was used. The profile traversed the domain 200 times to highlight the difference.

Figure 1. Comparison of the Gaussian profiles for the DG, FD, and CD schemes

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