A discontinuous Galerkin residual‐based variational multiscale method for modeling subgrid‐scale behavior of the viscous Burgers equation

Abstract

Summary We initiate the study of the discontinuous Galerkin residual‐based variational multiscale (DG‐RVMS) method for incorporating subgrid‐scale behavior into the finite element solution of hyperbolic problems. We use the one‐dimensional viscous Burgers equation as a model problem, as its energy dissipation mechanism is analogous to that of turbulent flows. We first develop the DG‐RVMS formulation for a general class of nonlinear hyperbolic problems with a diffusion term, based on the decomposition of the true solution into discontinuous coarse‐scale and fine‐scale components. In contrast to existing continuous variational multiscale methods, the DG‐RVMS formulation leads to additional fine‐scale element interface terms. For the Burgers equation, we devise suitable models for all fine‐scale terms that do not use ad hoc devices such as eddy viscosities but instead directly follow from the nature of the fine‐scale solution. In comparison to single‐scale discontinuous Galerkin methods, the resulting DG‐RVMS formulation significantly reduces the energy error of the Burgers solution, demonstrating its ability to incorporate subgrid‐scale behavior in the discrete coarse‐scale system.

Unification of variational multiscale analysis and Nitsche's method, and a resulting boundary layer fine-scale model

“…( 8). However, the particular structures of the coarse-scale problem and the Nitsche projector allow for a more direct inversion of (part of) the fine scales in the coarse-scale equation [33,34]. First, we recognize that the fine-scale terms in eq.…”

“…Additionally, the fine-scale solution does, by design, not vanish on the Dirichlet boundary, which violates one of the key assumptions on which traditional residual-based fine-scale models are built. In previous work, we focused on discontinuous Galerkin methods, where the discontinuities between elements give rise to similar issues [32][33][34]. The goal of this article is to completely eliminate these issues for weak boundary imposition in Nitsche-type formulations.…”