Implicit Solution of the Unsteady Euler Equations for High-Order Accurate Discontinuous Galerkin Discretizations

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.