Pseudo-time stepping methods for space-time discontinuous Galerkin discretizations of the compressible Navier-Stokes equations

The space-time discontinuous Galerkin discretization of the compressible Navier-Stokes equations results in a non-linear system of algebraic equations, which we solve with pseudo-time stepping methods. We show that explicit Runge-Kutta methods developed for the Euler equations suffer from a severe stability constraint linked to the viscous part of the equations and propose an alternative to relieve this constraint while preserving locality. To evaluate its effectiveness, we compare with an implicit-explicit Runge-Kutta method which does not suffer from the viscous stability constraint. We analyze the stability of the methods and illustrate their performance by computing the flow around a 2D airfoil and a 3D delta wing at low and moderate Reynolds numbers.

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

[2]  Chi-Wang Shu,et al.  The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems , 1998 .

[3]  Alexander Ostermann,et al.  Stability of W-methods with applications to operator splitting and to geometric theory , 2002 .

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

[5]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[6]  Gesellschaft für Angewandte Mathematik und Mechanik,et al.  Numerical simulation of compressible Navier-Stokes flows : a GAMM-Workshop , 1987 .

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

[8]  William L. Kleb,et al.  Efficient .Multi-Stage Time Marching for Viscous' Flows via. Local Preconditioning , 1999 .

[9]  Vít Dolejší,et al.  On the discontinuous Galerkin method for the numerical solution of the Navier–Stokes equations , 2004 .

[10]  Mark D. Sanetrik,et al.  Time-accurate Navier-Stokes calculations with multigrid acceleration , 1993 .

[11]  Jacques Periaux,et al.  Numerical Simulation of Compressible Navier-Stokes Flows , 1987 .

[12]  H. van der Ven,et al.  h-Multigrid for space-time discontinuous Galerkin discretizations of the compressible Navier-Stokes equations , 2007, J. Comput. Phys..

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

[14]  J. C. Kok,et al.  An Industrially applicable solver for compressible, turbulente flows , 1998 .

[15]  R. Hartmann,et al.  Symmetric Interior Penalty DG Methods for the CompressibleNavier-Stokes Equations I: Method Formulation , 2005 .

[16]  P. Wesseling.  Principles of Computational Fluid Dynamics , 2000 .

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

[18]  J. Tinsley Oden,et al.  A discontinuous hp finite element method for the Euler and Navier–Stokes equations , 1999 .

[19]  S. Rebay,et al.  A High-Order Accurate Discontinuous Finite Element Method for the Numerical Solution of the Compressible Navier-Stokes Equations , 1997 .

[20]  J. V. D. Vegt,et al.  Space--time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows: I. general formulation , 2002 .

[21]  H. Vanderven.  Space?time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flowsII. Efficient flux quadrature , 2002 .

[22]  Jaap J. W. van der Vegt,et al.  Space-Time Discontinuous Galerkin Method for the Compressible Navier-Stokes , 2006 .

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

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

[25]  J. V. D. Vegt,et al.  Space-time discontinuous Galerkin method for advection-diffusion problems on time-dependent domains , 2006 .