A dissipative Filter for the Discontinuous Galerkin method

Abstract

Complex three dimensional flows in domains with nontrivial geometry require use of mixed type unstructured meshes. The discontinuous Galerkin (DG) method is an unstructured grid method and development of a unified approach for discontinuity capturing with the DG method when higher order expansions are employed is therefore required. A systematic procedure for incorporating a dissipative mechanism to DG discretizations in unstructured meshes is developed. This adaptive non-linear dissipative mechanism enables capturing discontinuities and steep flow gradients without numerical oscillations. A filter designed and extensively used in the development of low dissipative well-balanced high order accurate finite difference schemes is employed. The filtering operator is adapted to the finite element context of DG discretizations. The filter operator is constructed and applied in the canonical computational domain where it can be applied in a direction per direction basis and then extended for all element types in unstructured meshes using collapsed coordinate transformations. The performance of the proposed dissipative mechanism is demonstrated and evaluated for different orders of expansions for one-dimensional and multidimensional problems with exact solutions. It is further applied for a number of standard inviscid flow test problems including strong shocks interactions to demonstrate the potential of the method. The dissipative mechanism yields results comparable to three-dimensional TVB and hierarchical limiting. Furthermore, the proposed approach is suitable for h/p-adaptivity in order to locally enhance resolution for three dimensional flow simulations that include discontinuities and complex flow features.

A local mesh refinement approach for large‐eddy simulations of turbulent flows

“…Grid‐based (as opposed to meshless) numerical solution of the governing equations of fluid flow can be achieved on structured or unstructured grids. Although the recent trend in LES has been for the application of finite volume and finite element methods on unstructured grids, the present study focuses on the use of structured grids for two principal reasons: firstly, the accuracy of the computation on unstructured grids often suffers when extreme gradients are experienced along the dominant flow direction, as is the case in boundary layer flow, for example , and secondly, unstructured grids are often not suitable for use with multigrid methods, which have been used by the authors in the past to achieve very efficient computations. In addition, the immersed boundary method (IBM) permits the accurate and efficient representation of complex solid bodies on fixed Cartesian grids, thereby circumventing one of the main perceived limitations of structured grids, which is their inability to represent complex geometries.…”