Implicit solution of the unsteady Euler equations for high-order accurate discontinuous Galerkin discretizations

  • Authors:

  • Li Wang;Dimitri J. Mavriplis

  • Affiliations:

  • Department of Mechanical Engineering, University of Wyoming, Laramie, WY 82071-3295, United States;Department of Mechanical Engineering, University of Wyoming, Laramie, WY 82071-3295, United States

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2007

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Abstract

Efficient solution techniques for high-order accurate time-dependent problems are investigated for solving the two-dimensional non-linear Euler equations in this work. The spatial discretization consists of a high-order accurate discontinuous Galerkin (DG) approach. Implicit time-integration techniques are considered exclusively in order to avoid the stability restrictions of explicit methods. Standard backwards differencing methods (BDF1 and BDF2) as well as a second-order Crank-Nicholson (CN2) and a fourth-order implicit Runge-Kutta (IRK4) scheme are considered in an attempt to balance the spatial and temporal accuracy of the overall approach. The implicit system arising at each time step is solved using a p-multigrid approach, which is shown to produce h independent convergence rates, while remaining relatively insensitive to the time-step size. The Crank-Nicholson methodology, although not L-stable, demonstrates superior performance compared to the BDF2 scheme for the problems chosen in this work. However, the fourth-order accurate implicit Runge-Kutta scheme is found to be the most efficient in terms of computational cost for a given accuracy level as compared to the lower-order schemes, in spite of the added cost per time step, and the benefits of this scheme increase for tighter error tolerances.