A discontinuous Galerkin residual‐based variational multiscale method for modeling subgrid‐scale behavior of the viscous Burgers equation. (19th July 2018)

• Record Type: Journal Article
• Authors: Stoter, Stein K.F.; Turteltaub, Sergio R.; Hulshoff, Steven J.; Schillinger, Dominik
• Is Part Of: International journal for numerical methods in fluids. Volume 88:Number 5(2018)
• Page Start: 217
• Page End: 238
• Publication Date: 2018-07-19
• Subjects: Burgers turbulence -- discontinuous Galerkin methods -- residual‐based multiscale modeling -- variational multiscale method; Fluid dynamics -- Mathematics -- Periodicals; 532
• DOI: 10.1002/fld.4662
• Languages: English
• ISSNs: 0271-2091

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. Abstract : In this paper, we propose a new residual‐based variational multiscale subgrid‐scale model that is suitable for discontinuous Galerkin implementations. We use the jumps of the discontinuous basis functions as a measure for the fine‐scale element boundary values. Then, we incorporate the homogenized effect of these nonvanishing fine‐scale boundary values onto the element interior. We investigate the effect of our model for a viscous Burgers equation, where we focus on the accuracy of the energy spectra.