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A discontinuous Galerkin residual‐based variational multiscale method for modeling subgrid‐scale behavior of the viscous Burgers equation
Author(s) -
Stoter Stein K.F.,
Turteltaub Sergio R.,
Hulshoff Steven J.,
Schillinger Dominik
Publication year - 2018
Publication title -
international journal for numerical methods in fluids
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.938
H-Index - 112
eISSN - 1097-0363
pISSN - 0271-2091
DOI - 10.1002/fld.4662
Subject(s) - burgers' equation , discontinuous galerkin method , mathematics , scale (ratio) , dissipation , galerkin method , mathematical analysis , residual , nonlinear system , finite element method , hyperbolic partial differential equation , partial differential equation , physics , algorithm , quantum mechanics , thermodynamics
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.