On the efficiency of a matrix-free linearly implicit time integration strategy for high-order Discontinuous Galerkin solutions of incompressible turbulent flows

The paper aims at investigating and comparing the efficiency of matrix-based and matrix-free implementations of implicit time integration strategies, here linearly implicit Rosenbrock-type Runge-Kutta schemes, in the context of a high order Discontinous Galerkin space discretization applied to DNS/ILES of turbulent flows. In particular, the effects of time step size, GMRES solution tolerance and preconditioner, mesh size, order of polynomial approximation of the solution and parallel efficiency are here considered. The performance of the solvers have been compared though the solution of the two-dimensional laminar traveling waves test case, the ILES of the Rayleigh-Bénard natural convection up to Ra=108, and the channel flow up to Reτ=950. The results highlight that while matrix-based strategies are the most adequate for low-order polynomial approximations, considerable improvements in computational efficiency can be achieved, both in terms of CPU time and memory footprint, by a proper use of the matrix-free solver for three-dimensional test cases and very high-order accurate DG space discretizations.