High order Finite Difference/Discontinuous Galerkin schemes for the incompressible Navier-Stokes equations with implicit viscosity

In this work we propose a novel numerical method for the solution of the incompressible Navier-Stokes equations on Cartesian meshes in 3D. The semi-discrete scheme is based on an explicit discretization of the nonlinear convective flux tensor and an implicit treatment of the pressure gradient and viscous terms. In this way, the momentum equation is formally substituted into the divergence-free constraint, thus obtaining an elliptic equation on the pressure which eventually maintains at the discrete level the involution on the divergence of the velocity field imposed by the governing equations. This makes our method belonging to the class of so-called structure-preserving schemes. High order of accuracy in space is achieved using an efficient CWENO reconstruction operator that is exploited to devise a conservative finite difference scheme for the convective terms. Implicit central finite differences are used to remove the numerical dissipation in the pressure gradient discretization. To avoid the severe time step limitation induced by the viscous eigenvalues related to the parabolic terms in the governing equations, we propose to devise an implicit local discontinuous Galerkin (DG) solver. The resulting viscous sub-system is symmetric and positive definite, therefore it can be efficiently solved at the aid of a matrix-free conjugate gradient method. High order in time is granted by a semi-implicit IMEX time stepping technique. Convergence rates up to third order of accuracy in space and time are proven, and a suite of academic benchmarks is shown in order to demonstrate the robustness and the validity of the novel schemes, especially in the context of high viscosity coefficients.