High-order linearly implicit two-step peer schemes for the discontinuous Galerkin solution of the incompressible Navier–Stokes equations

In this work the use of high-order linearly implicit Rosenbrock-type two-step peer schemes has been investigated to integrate in time the high-order discontinuous Galerkin space discretization of the incompressible Navier–Stokes equations. The aim of the present paper is (i) to describe the implementation of the schemes in the DG code MIGALE with focus on the computation of the set of the coefficients and the starting procedure, (ii) to describe the coupling of the scheme with an adaptive time-step strategy in order to investigate its effect on the robustness and computational efficiency of the simulations, and (iii) to provide some practical informations regarding the choice of the “optimal” time integration for LES and DNS on the basis of the requested accuracy. Peer schemes, up to sixth order, have been considered and compared with traditional one-step linearly implicit Rosenbrock, up to fifth order, and ESDIRK, up to fourth order, schemes available in literature in terms of accuracy and computational efficiency. For the sake of completeness, the sets of coefficients of the schemes here considered have been reported in an appendix. The reliability, robustness and accuracy of the proposed implementation have been assessed by computing the Prothero–Robinson example, the laminar travelling waves solution on a doubly periodic unit square domain and the laminar flow around a circular cylinder for a Reynolds number Re=100. Travelling waves and cylinder testcases have been also modified to investigate the behaviour of the schemes with time-dependent boundary conditions. In the former case replacing periodic boundary conditions with given boundary condition based on the analytical solution, while in the latter case considering a rotating cylinder.