A hybridized discontinuous Galerkin method on mapped deforming domains

Abstract 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.

[1]  Krzysztof J. Fidkowski,et al.  Error Estimation and Adaptation in Hybridized Discontinuous Galerkin Methods , 2014 .

[2]  Krzysztof J. Fidkowski,et al.  Output-based space-time mesh adaptation for the compressible Navier-Stokes equations , 2011, J. Comput. Phys..

[3]  Krzysztof J. Fidkowski,et al.  An Output-Based Dynamic Order Refinement Strategy for Unsteady Aerodynamics , 2012 .

[4]  R. Hartmann,et al.  Adaptive discontinuous Galerkin finite element methods for the compressible Euler equations , 2002 .

[5]  Rolf Rannacher,et al.  An optimal control approach to a posteriori error estimation in finite element methods , 2001, Acta Numerica.

[6]  P. Roe.  Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes , 1997 .

[7]  Bernardo Cockburn,et al.  Hybridizable Discontinuous Galerkin Methods , 2011 .

[8]  Endre Süli,et al.  DISCONTINUOUS GALERKIN METHODS FOR FIRST-ORDER HYPERBOLIC PROBLEMS , 2004 .

[9]  Bernardo Cockburn,et al.  An Embedded Discontinuous Galerkin Method for the Compressible Euler and Navier-Stokes Equations , 2011 .

[10]  Krzysztof J. Fidkowski,et al.  Output-based mesh adaptation for high order Navier-Stokes simulations on deformable domains , 2013, J. Comput. Phys..

[11]  Michael Woopen,et al.  A Comparison of Hybridized and Standard DG Methods for Target-Based hp-Adaptive Simulation of Compressible Flow , 2013, ArXiv.

[12]  Frédéric Alauzet,et al.  Time accurate anisotropic goal-oriented mesh adaptation for unsteady flows , 2012, J. Comput. Phys..

[13]  Krzysztof J. Fidkowski,et al.  A local sampling approach to anisotropic metric-based mesh optimization , 2016 .

[14]  Chi-Wang Shu,et al.  Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems , 2001, J. Sci. Comput..

[15]  Jaime Peraire,et al.  Discontinuous Galerkin Solution of the Navier-Stokes Equations on Deformable Domains , 2007 .

[16]  Raytcho D. Lazarov,et al.  Unified Hybridization of Discontinuous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order Elliptic Problems , 2009, SIAM J. Numer. Anal..

[17]  Michael J. Aftosmis,et al.  Adjoint Error Estimation and Adaptive Refinement for Embedded-Boundary Cartesian Meshes , 2007 .

[18]  Anthony T. Patera,et al.  "Natural norm" a posteriori error estimators for reduced basis approximations , 2006, J. Comput. Phys..

[19]  Bernardo Cockburn,et al.  An implicit high-order hybridizable discontinuous Galerkin method for nonlinear convection-diffusion equations , 2009, J. Comput. Phys..

[20]  Dimitri J. Mavriplis,et al.  Discrete Adjoint Based Adaptive Error Control in Unsteady Flow Problems , 2012 .

[21]  Sander Rhebergen,et al.  A space-time hybridizable discontinuous Galerkin method for incompressible flows on deforming domains , 2012, J. Comput. Phys..

[22]  Dimitri J. Mavriplis,et al.  Error estimation and adaptation for functional outputs in time-dependent flow problems , 2009, Journal of Computational Physics.

[23]  Bernardo Cockburn,et al.  An implicit high-order hybridizable discontinuous Galerkin method for linear convection-diffusion equations , 2009, Journal of Computational Physics.

[24]  Prabhu Ramachandran,et al.  Approximate Riemann solvers for the Godunov SPH (GSPH) , 2014, J. Comput. Phys..

[25]  J. Oden,et al.  hp-Version discontinuous Galerkin methods for hyperbolic conservation laws , 1996 .

[26]  Michael B. Giles,et al.  Adjoint Recovery of Superconvergent Functionals from PDE Approximations , 2000, SIAM Rev..

[27]  Dimitri J. Mavriplis,et al.  Error estimation and adaptation for functional outputs in time-dependent flow problems , 2009, J. Comput. Phys..

[28]  Haihang You,et al.  Adaptive Discontinuous Galerkin Finite Element Methods , 2009 .

[29]  James Lu,et al.  An a posteriori Error Control Framework for Adaptive Precision Optimization using Discontinuous Galerkin Finite Element Method , 2005 .

[30]  D. Darmofal,et al.  Review of Output-Based Error Estimation and Mesh Adaptation in Computational Fluid Dynamics , 2011 .

[31]  W. H. Reed,et al.  Triangular mesh methods for the neutron transport equation , 1973 .

[32]  Sander Rhebergen,et al.  Space-Time Hybridizable Discontinuous Galerkin Method for the Advection–Diffusion Equation on Moving and Deforming Meshes , 2013 .

[33]  S. Rebay,et al.  GMRES Discontinuous Galerkin Solution of the Compressible Navier-Stokes Equations , 2000 .

[34]  Boris Vexler,et al.  Adaptivity with Dynamic Meshes for Space-Time Finite Element Discretizations of Parabolic Equations , 2007, SIAM J. Sci. Comput..

[35]  Rolf Rannacher,et al.  Goal‐oriented space–time adaptivity in the finite element Galerkin method for the computation of nonstationary incompressible flow , 2012 .

[36]  S. Rebay,et al.  Numerical evaluation of two discontinuous Galerkin methods for the compressible Navier–Stokes equations , 2002 .

[37]  Bo Dong,et al.  A Hybridizable Discontinuous Galerkin Method for Steady-State Convection-Diffusion-Reaction Problems , 2009, SIAM J. Sci. Comput..

[38]  D. Venditti,et al.  Anisotropic grid adaptation for functional outputs: application to two-dimensional viscous flows , 2003 .

[39]  Timothy J. Barth,et al.  Space-Time Error Representation and Estimation in Navier-Stokes Calculations , 2013 .

[40]  David L. Darmofal,et al.  p-Multigrid solution of high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations , 2005 .