Matrix-free modified extended BDF applied to the discontinuous Galerkin solution of unsteady compressible viscous flows

In this work, a time-accurate integration of the discontinuous Galerkin space-discretized Navier-Stokes equations is performed exploiting the matrix-free (MF) approach to speed up the solution process of the modified extended backward differentiation formulae (MEBDF) schemes. MEBDF are high-order accurate implicit multistep schemes composed by three nonlinear stages. The proposed algorithm consists in solving the resulting nonlinear system of each stage with a preconditioned MF Newton/Krylov method using a frozen preconditioner strategy to improve its efficiency. Numerical results for compressible inviscid and viscous test cases, both with a known analytical solution, aim at assessing the performance of the proposed MF-MEBDF algorithm, by comparing it with the one obtained by using its matrix-explicit counterpart or the explicit strong stability preserving Runge-Kutta scheme. In particular, the influence of some relevant physical (low-speed flows) and discretization (aspect ratio, polynomial degree) aspects on the performance of the different time integration schemes are investigated, highlighting the pros and cons of the proposed algorithm and its effectiveness in solving nonstiff and stiff systems. Furthermore, the scalability in parallel computations of the proposed algorithm is investigated and, finally, its potential for efficient long-time simulations is demonstrated computing a laminar vortex shedding behind a circular cylinder at different Reynolds numbers and by comparing its effectiveness with that of the strong stability preserving Runge-Kutta scheme.