On the efficiency of implicit discontinuous Galerkin spectral element methods for the unsteady compressible Navier-Stokes equations

In this work we investigate the implicit time integration for the Discontinuous Galerkin Spectral Element Method (DGSEM) with respect to its efficiency and accuracy compared to explicit time integration. Considering unsteady simulations of laminar and turbulent flows described by the compressible Navier-Stokes equations, not only the spatial but also the temporal discretization is of high importance. Due to the explicit time step restriction caused by the stability condition, explicit schemes are widely used for time accurate simulations. In the case of very stiff problems this condition becomes severe leading to the consideration of implicit time integration schemes, which can be constructed as unconditionally stable. The implicit time step is contrastingly only driven by accuracy requirements. However, solving the arising (non-) linear equation systems within the DGSEM context is still a challenging issue. We consider Explicit first stage Singly Diagonally Implicit Runge-Kutta (ESDIRK) schemes in combination with a Jacobian-Free Newton-Krylov solver. In the first part, we introduce a novel strategy for solving the non-linear and linear systems with adaptive tolerances in order to avoid over- and under-solving. These tolerances are automatically adjusted to temporal accuracy requirements. This strategy leads not only to an user friendly handling but also to an highly efficient solver. In a second step, we employ the block-Jacobi preconditioner neglecting all the off-diagonals blocks, so that the resulting linear system is able to be solved element-locally. The advantages of this preconditioner are low storage requirements, parallel scalability and simple implementation. The results highlight that implicit DGSEM can be competitive with an explicit Runge-Kutta scheme in terms of computational time and accuracy. The implicit solver can outperform the explicit scheme in the case of very severe time step restrictions with the same spatial accuracy