Discontinuous Galerkin methods and stability analysis

Abstract: During recent years, discontinuous Galerkin (DG) high order methods (order ≥ 3) have gained popularity to compute accurate solutions of the Navier-Stokes equations. The DG technique is characterised by low numerical errors (i.e. dispersion and diffusion) and its ability to perform mesh refinement (increased number of mesh nodes or h-refinement) and/or polynomial enrichment (p-refinement) to achieve accurate solutions. DG methods are an extension of continuous spectral methods where the continuity constraint required on edge boundaries is relaxed, allowing for discontinuities in the numerical solution. In this talk, we present applications of discontinuous Galerkin techniques for compressible and incompressible flows. Examples include laminar and turbulent flows over bluff and slender geometries and rotating wind turbines modelled using high order sliding meshes. Additionally, local anisotropic p-adaption and multigrid strategies for DG methods are presented. In the second part of the talk, we use high order solvers (e.g. discontinuous Galerkin) to perform linear stability analysis and sensitivity analysis. Adjoint-based sensitivity analysis builds upon linear stability and provides information of the most sensitive flow regions that may be modified to control particular flow features. Examples include 3D cavities, turbulent transonic flows and wind turbines.