A Non-Dissipative, Energy-Conserving, Arbitrary High-Order Numerical Method and Its Efficient Implementation for Incompressible Flow Simulation in Complex Geometries

In the inviscid limit, the energy of a velocity field satisfying the incompressible Navier–Stokes equations is conserved. Non-dissipative numerical methods that discretely mimic this energy conservation feature have been demonstrated in the literature to be extremely valuable for robust and accurate large-eddy simulations of high Reynolds number incompressible turbulent flows. For complex geometries, such numerical methods have been traditionally developed using the finite volume framework and they have been at best second-order accurate. This paper proposes a non-dissipative and energy-conserving numerical method that is arbitrary high-order accurate for triangle/tetrahedral meshes along with its efficient implementation. The proposed method is a Hybridizable Discontinuous Galerkin (HDG) method. The crucial ingredients of the numerical method that lead to the discretely non-dissipative and energy-conserving features are: (i) The tangential velocity on the interior faces, just for the convective term, is set using the non-dissipative central scheme and the normal velocity is enforced to be continuous, that is, $H$(div)-conforming. (ii) An exactly (pointwise) divergence-free basis is used in each element of the mesh for the stability of the convective discretization. (iii) The combination of velocity, pressure, and velocity gradient spaces is carefully chosen to avoid using stabilization which would introduce numerical dissipation. The implementation description details our choice of the orthonormal and degree-ordered basis for each quantity and the efficient local and global problem solution using them. Numerical experiments demonstrating the various features of the proposed method are presented. The features of this HDG method make it ideal for high-order LES of incompressible flows in complex geometries.