Residual-Based Variational Multiscale Modeling in a Discontinuous Galerkin Framework

Stoter, Stein K. F.; Turteltaub, Sergio; Hulshoff, Steven J.; Schillinger, Dominik

doi:10.1137/17m1147044

Cited by 6 publications

(9 citation statements)

References 49 publications

(103 reference statements)

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“…The multiscale formulation thus opens the door for a new perspective on DG methods and their numerical properties. In our previous work, this was demonstrated for the one‐dimensional advection‐diffusion problem, where the use of upwind numerical fluxes was shown to be interpretable as an ad hoc remedy for missing volumetric fine‐scale terms.…”

Section: Introductionmentioning

confidence: 90%

“…In this section, we extend the DG‐RVMS method, introduced in our preliminary work, to nonlinear hyperbolic problems with a viscous term. For a periodic domain Ω, this class of boundary value problems is defined as

({, \begin{matrix} array u_{t} - ν Δ u + \nabla \cdot f (u) = g (x, t), & array in Ω \times (t^{0}, T] \\ array u = u (t^{0}), & array on Ω \times {t^{0}}, \end{matrix})

where f ( u ) is a (potentially nonlinear) flux function.…”

Section: Vms Formulation In a Discontinuous Approximation Spacementioning

confidence: 99%

“…In this section, we extend the DG-RVMS method, introduced in our preliminary work, 33 to nonlinear hyperbolic problems with a viscous term. For a periodic domain Ω, this class of boundary value problems is defined as…”

Section: Vms Formulation In a Discontinuous Approximation Spacementioning

confidence: 99%

“…The first is a fine‐scale volumetric term, which is formulated in terms of a residual‐based model that also takes into account the nonhomogeneous fine‐scale element boundary values. The second are independent fine‐scale terms at element interfaces, which are formulated in terms of a new fine‐scale “interface model.” In a preliminary work, we demonstrated for the one‐dimensional Poisson problem that existing DG formulations, such as the symmetric interior penalty (IP) method, can be rederived by choosing particular fine‐scale interface models. The multiscale formulation thus opens the door for a new perspective on DG methods and their numerical properties.…”

Section: Introductionmentioning

confidence: 99%

“…To some extent, this may be attributed to the importance of coarse-scale continuity in the derivation of the VMS method. 2,10,32 The discontinuous Galerkin residual-based variational multiscale (DG-RVMS) method that we have presented recently in a preliminary work 33 no longer relies on the level of continuity of the coarse-scale function space. On the basis of the decomposition of the true solution into discontinuous coarse-scale and fine-scale components, it features two types of fine-scale contributions.…”

Section: Introductionmentioning

confidence: 99%

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

Stoter

Turteltaub

Hulshoff

et al. 2018

Numerical Methods in Fluids

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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.

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Section: Introductionmentioning

confidence: 90%