On the dissipation of H (div) -conforming schemes for incompressible flows

In this paper, we present a systematic construction of a H (div)-conforming numerical dissipation for time-dependent incompressible Euler and Navier-Stokes equations. The goal is to improve the performance of the central flux scheme. The construction is a generalization of the upwind flux scheme from the dissipation point of view and makes use of the discontinuity of numerical quantities across interior edges. It is physically connected to the implicit large eddy simulation used in turbulent flow simulations. Examples are constructed when the jump of the gradient or curl of numerical velocity is used, and their performance is tested through numerical experiments. Numerical results show that the added dissipation does a good job in reducing numerical errors and in preserving the right physics.