A hybridized discontinuous Galerkin method on mapped deforming domains

• Published in: Computers & fluids Vol. 139; no. C; pp. 80 - 91
• Main Author: Fidkowski, Krzysztof J.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Ltd 05.11.2016; Elsevier BV; Elsevier
• Subjects: Advection; Compressibility; Computer simulation; Convection; Convergence; Deformation; Deforming domains; Diffusion rate; Discretization; Economic models; Euler equations; Finite element method; Fluid dynamics; Fluid flow; Galerkin method; High order finite elements; Hybridized discontinuous Galerkin; Mapping; Mathematical analysis; Mesh motion; Navier-Stokes equations; Stabilization; Stokes law (fluid mechanics); Transformations (mathematics); Deforming domains; Hybridized discontinuous Galerkin; High order finite elements; Mesh motion
• ISSN: 0045-7930; 1879-0747
• DOI: 10.1016/j.compfluid.2016.04.004

•Hybridized DG can be extended to ALE formulations.•The HDG gradient equation acquires a source term in the ALE mapping.•Least-squares solution initialization is required for optimal output error convergence rates.•Choice of HDG viscous stabilization is most important for diffusion-dominated flows.•Steady-state adapted meshes may perform poorly for unsteady simulations. In this paper we present a hybridized discontinuous Galerkin (HDG) discretization for unsteady simulations of convection-dominated flows on mapped deforming domains. Mesh deformation is achieved through an arbitrary Lagrangian-Eulerian transformation with an analytical mapping. We present details of this transformation applied to the HDG system of equations, with focus on the auxiliary gradient equation, viscous stabilization, and output calculation. We discuss conditions under which optimal unsteady output convergence rates can be attained, and we show that both HDG and discontinuous Galerkin (DG) achieve these rates in advection-dominated flows. Results for scalar advection-diffusion and the Euler equations verify the implementation of the mesh motion mapping for both discretizations, show that HDG and DG yield similar results on a given mesh, and demonstrate differences in output convergence rates depending on the choice of HDG viscous stabilization. We note that such similar results bode well for HDG, which has fewer globally-coupled degrees of freedom compared to DG. A simulation of the unsteady compressible Navier-Stokes equations demonstrates again very similar results for HDG and DG and illustrates a pitfall of using steady-state adapted meshes for accurate unsteady simulations.