An implicit matrix-free Discontinuous Galerkin solver for viscous and turbulent aerodynamic simulations

This paper presents some recent advancements of the computational efficiency of a Discontinuous Galerkin (DG) solver for the Navier–Stokes (NS) and Reynolds Averaged Navier Stokes (RANS) equations. The implementation and the performance of a Newton–Krylov matrix-free (MF) method is presented and compared with the matrix based (MB) counterpart. Moreover two solution strategies, developed in order to increase the solver efficiency, are discussed and experimented. Numerical results of some test cases proposed within the EU ADIGMA (Adaptive Higher-Order Variational Methods for Aerodynamic Applications in Industry) project demonstrate the capabilities of the method.

[1]  G. Bergeles,et al.  Notes on Numerical Fluid Mechanics and Multidisciplinary Design , 2012 .

[2]  Andrea Crivellini,et al.  High-order discontinuous Galerkin discretization of transonic turbulent flows , 2009 .

[3]  Sanjay Mittal,et al.  Parallel finite element computation of incompressible flows , 2009, Parallel Comput..

[4]  Douglas N. Arnold,et al.  Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[5]  P. Tesini,et al.  High‐order accurate p‐multigrid discontinuous Galerkin solution of the Euler equations , 2009 .

[6]  Alessandro Colombo,et al.  Robust and Efficient Implementation of Very High-Order Discontinuous Galerkin Methods in CFD , 2010 .

[7]  Marek Behr,et al.  Implementation of implicit finite element methods for incompressible flows on the CM-5 , 1994 .

[8]  Ewald Krämer,et al.  A parallel, high-order discontinuous Galerkin code for laminar and turbulent flows , 2008 .

[9]  Bernardo Cockburn.  Discontinuous Galerkin methods , 2003 .

[10]  Christian B Allen,et al.  48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition , 2010 .

[11]  Dimitri J. Mavriplis,et al.  High-order discontinuous Galerkin methods using an hp-multigrid approach , 2006, J. Comput. Phys..

[12]  Analysis of a hybrid p-Multigrid method for the discontinuous Galerkin discretisation of the Euler equations , 2006 .

[13]  Jean-François Remacle,et al.  Hierarchic multigrid iteration strategy for the discontinuous Galerkin solution of the steady Euler equations , 2006 .

[14]  Dimitri J. Mavriplis,et al.  Multigrid algorithms for high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations , 2009, J. Comput. Phys..

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

[16]  D. Keyes,et al.  Jacobian-free Newton-Krylov methods: a survey of approaches and applications , 2004 .

[17]  S. Rebay,et al.  High-Order Accurate Discontinuous Finite Element Solution of the 2D Euler Equations , 1997 .

[18]  David W. Zingg,et al.  A Jacobian-free Newton-Krylov algorithm for compressible turbulent fluid flows , 2009, J. Comput. Phys..

[19]  Francesco Bassi,et al.  A High Order Discontinuous Galerkin Method for Compressible Turbulent Flows , 2000 .

[20]  Homer F. Walker,et al.  NITSOL: A Newton Iterative Solver for Nonlinear Systems , 1998, SIAM J. Sci. Comput..

[21]  Laslo T. Diosady,et al.  Preconditioning methods for discontinuous Galerkin solutions of the Navier-Stokes equations , 2009, J. Comput. Phys..

[22]  R. Schwane,et al.  ON THE ACCURACY OF UPWIND SCHEMES FOR THE SOLUTION OF THE NAVIER-STOKES EQUATIONS , 1987 .

[23]  F. Brezzi,et al.  Discontinuous Galerkin approximations for elliptic problems , 2000 .

[24]  Allen D. Malony,et al.  The Tau Parallel Performance System , 2006, Int. J. High Perform. Comput. Appl..

[25]  Rainald Löhner,et al.  A p-multigrid discontinuous Galerkin method for the Euler equations on unstructured grids , 2006 .

[26]  Alessandro Colombo,et al.  Very High-Order Accurate Discontinuous Galerkin Computation of Transonic Turbulent Flows on Aeronautical Configurations , 2010 .

[27]  Marco Luciano Savini,et al.  Discontinuous Galerkin solution of the Reynolds-averaged Navier–Stokes and k–ω turbulence model equations , 2005 .

[28]  Per-Olof Persson,et al.  Newton-GMRES Preconditioning for Discontinuous Galerkin Discretizations of the Navier--Stokes Equations , 2008, SIAM J. Sci. Comput..

[29]  M. Y. Hussaini,et al.  An efficient implicit discontinuous spectral Galerkin method , 2001 .