Legendre Pseudospectral Viscosity Method for Nonlinear Conservation Laws

A Spectral Vanishing Viscosity Method for Large-Eddy Simulations

“…The one-dimensional version has already been extended to collocation methods [32], and the fact that the SVV is a second-order operator allows a straightforward implementation in finite element codes unlike hyperviscous kernels. On the other hand, at present it is not clear how to develop rigorously the SVV method for finite difference methods avoiding the current empiricism, although some progress on this front has been made by employing similar ideas [47].…”

“…Comparing the Fourier analog of this eddy viscosity employed in LES [34] to the viscosity kernel This range has been used in most of the numerical experiments so far (see, for example, [27,32]) and is consistent with the theoretical results [22]. In the plot it is shown that, in general, the two forms of viscosity have similar distributions, but the SVV form does not affect the first one-third or one-half of the spectrum (viscosity-free portion) and it increases faster than the Kraichnan/Chollet-Lesieur eddy viscosity in the higher wavenumber range, e.g., in the second-half of the spectrum.…”

“…In subsequent work, however, a smooth kernel was used, since it was found that the C ∞ smoothness ofQ k improves the resolution of the SVV method. For Legendre pseudo-spectral methods, Maday et al [32] used ≈ N −1 , activated for modes…”

Mathematical analysis of a spectral hyperviscosity LES model for the simulation of turbulent flows

“…We choose to construct the kernel in such a way that it acts only on the high wave numbers of the velocity field, namely N i ≤ |k| ∞ ≤ N , N i standing for an intermediate cutoff wave number. This idea is quite similar to the spectral viscosity technique that Tadmor [5,25,30] developed for nonlinear scalar conservation laws.…”

“…On the other hand, LES methods based on spectral approximations generally fall short of complying with these two criteria. The spectral viscosity model of Tadmor [5,25,30], which was originally developed for nonlinear scalar conservation laws, and later extended to large eddy simulation of viscous flows by Karamanos and Karniadakis [13] or Adams and Stolz [1], may not yield a dissipative solution. Other techniques such as the spectral eddy-viscosity methods developed by Kraichnan [15], Chollet and Lesieur [7], Lamballais, Métais, and Lesieur [18], Lesieur and Roggalo [21], or McComb and Young [26], generally lose spectral accuracy when the flow is eventually resolved and does not guarantee convergence to a dissipative solution when the cutoff goes to infinity.…”