A discontinuous Galerkin method based on a Taylor basis for the compressible flows on arbitrary grids

A discontinuous Galerkin method based on a Taylor basis is presented for the solution of the compressible Euler equations on arbitrary grids. Unlike the traditional discontinuous Galerkin methods, where either standard Lagrange finite element or hierarchical node-based basis functions are used to represent numerical polynomial solutions in each element, this DG method represents the numerical polynomial solutions using a Taylor series expansion at the centroid of the cell. Consequently, this formulation is able to provide a unified framework, where both cell-centered and vertex-centered finite volume schemes can be viewed as special cases of this discontinuous Galerkin method by choosing reconstruction schemes to compute the derivatives, offer the insight why the DG methods are a better approach than the finite volume methods based on either TVD/MUSCL reconstruction or essentially non-oscillatory (ENO)/weighted essentially non-oscillatory (WENO) reconstruction, and has a number of distinct, desirable, and attractive features, which can be effectively used to address some of shortcomings of the DG methods. The developed method is used to compute a variety of both steady-state and time-accurate flow problems on arbitrary grids. The numerical results demonstrated the superior accuracy of this discontinuous Galerkin method in comparison with a second order finite volume method and a third-order WENO method, indicating its promise and potential to become not just a competitive but simply a superior approach than its finite volume and ENO/WENO counterparts for solving flow problems of scientific and industrial interest

Discontinuous subgrid formulations for transport problems

In this paper we develop two discontinuous Galerkin formulations within the framework of the two-scale subgrid method for solving advection–diffusion-reaction equations. We reformulate, using broken spaces, the nonlinear subgrid scale (NSGS) finite element model in which a nonlinear eddy viscosity term is introduced only to the subgrid scales of a finite element mesh. Here, two new subgrid formulations are built by introducing subgrid stabilized terms either at the element level or on the edges by means of the residual of the approximated resolved scale solution inside each element and the jump of the subgrid solution across interelement edges. The amount of subgrid viscosity is scaled by the resolved scale solution at the element level, yielding a self adaptive method so that no additional stabilization parameter is required. Numerical experiments are conducted in order to demonstrate the behavior of the proposed methodology in comparison with some discontinuous Galerkin methods.Indisponível