Longer time simulation of the unsteady NavierStokes equations based on a modified convective formulation
Abstract
For the discretization of the convective term in the NavierStokes equations (NSEs), the commonly used convective formulation (CONV) does not preserve the energy if the divergence constraint is only weakly enforced. In this paper, we apply the skewsymmetrization technique in [B. Cockburn, G. Kanschat and D. Schötzau, Math. Comp., 74 (2005), pp. 10671095] to conforming finite element methods, which restores energy conservation for CONV. The crucial idea is to replace the discrete advective velocity with its a $H(\operatorname{div})$conforming divergencefree approximation in CONV. We prove that the modified convective formulation also conserves linear momentum, helicity, 2D enstrophy and total vorticity under some appropriate senses. Its a Picardtype linearization form also conserves them. Under the assumption $\boldsymbol{u}\in L^{2}(0,T;\boldsymbol{W}^{1,\infty}(\Omega)),$ it can be shown that the Gronwall constant does not explicitly depend on the Reynolds number in the error estimates. The long time numerical simulations show that the linearized and modified convective formulation has a similar performance with the EMAC formulation and outperforms the usual skewsymmetric formulation (SKEW).
 Publication:

arXiv eprints
 Pub Date:
 December 2021
 arXiv:
 arXiv:2112.02330
 Bibcode:
 2021arXiv211202330L
 Keywords:

 Mathematics  Numerical Analysis