Variational multiscale stabilization of high-order spectral elements for the advection–diffusion equation

Efficient construction of unified continuous and discontinuous Galerkin formulations for the 3D Euler equations

“…To emphasize the need for stabilization of high-order Galerkin methods in operational mode, we compare results obtained with and without artificial viscosity; other advanced techniques for stabilization can be found in Marras et al [27]. Different implementations of spatial discretization schemes are compared by measuring the wall clock time taken to complete simulations.…”

A parameter-free dynamic alternative to hyper-viscosity for coupled transport equations: Application to the simulation of 3D squall lines using spectral elements

“…Because we already describe the spectral element approximation of the scalar counterpart of Eq. (4) in our previous work (Marras et al (2012)), we can omit the details on spectral elements and proceed to the description of their stabilization.…”

“…For up to cubic elements with equally spaced nodes, a derivation of the components of ⌧ was obtained by Marras et al (2012) although, for the sake of simplicity and without a great loss of accuracy, it is not used in this study. Because b( h , q h ) is a function of the equation residual and of a characteristic grid size, the e↵ect of stabilization goes to zero as the exact solution is approached.…”

“…To our knowledge, the only application of VMS to high order spectral elements for coupled problems appears in Gjesdal et al (2009) for the turbulence modeling of incompressible flows. In the case of Marras et al (2012), VMS+DC was only applied to solve one scalar advection problem and verified against standard benchmarks in one and two dimensions. The characteristic velocities of deep convection may be orders of magnitude larger than those in the purely academic tests presented in our previous work, thereby subjecting the scheme to a more demanding test when it comes to stabilization and treatment of over-and undershoots.…”

Localized Artificial Viscosity Stabilization of Discontinuous Galerkin Methods for Nonhydrostatic Mesoscale Atmospheric Modeling

“…These include, but are not limited to the streamline-upwind/Petrov-Galerkin (SUPG) type artificial viscosity [17,18,19,20], the Variational Multi-scale method (VMS) [21,22], localized artificial diffusivity using physical principles [10,11,23,24,25,26,27], the residual based artificial viscosity [14,15,28,29,30], the entropy artificial viscosity [16,31,32], the spectral vanishing viscosity [12,33], and the Laplacian artificial viscosity [13,34,35,36]. Other studies of the artificial viscosity methods can be found in References [37,38,39,40,41], just to name a few.…”