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 fourth-order implicit Runge-Kutta scheme (IRK4) 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 higher-order time integration schemes such as fourth-order implicit Runge-Kutta are found to be more efficient in terms of computational cost for a given accuracy level as compared to the lower order BDF1 and BDF2 schemes.