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.