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.
|Original language||English (US)|
|Number of pages||22|
|Journal||International Journal for Numerical Methods in Fluids|
|State||Published - Oct 20 2018|
Bibliographical noteFunding Information:
National Science Foundation, Grant/Award Number: 1651577
D. Schillinger gratefully acknowledges support from the National Science Foundation through NSF CAREER Award 1651577.
- Burgers turbulence
- discontinuous Galerkin methods
- residual-based multiscale modeling
- variational multiscale method