A stable discontinuous Galerkin method based on high‐order dual splitting scheme without additional stabilization term for incompressible flows

Abstract

AbstractIn this work, we consider equal‐order discontinuous Galerkin (DG) solver for incompressible Navier–Stokes equations based on high‐order dual splitting scheme. In order to stay stable, the time step size of this method has been reported that is strictly limited. The upper bound of time step size is restricted by Courant–Friedrichs–Lewy (CFL) condition (Hesthaven and Warburton, 2007) and lower bound is required to be larger than the critical value which depends on Reynolds number and spatial resolution (Ferrer et al., 2014). For high‐Reynolds‐number flow problems, if the spatial resolution is low, the critical value may be larger than CFL condition, then instability will occur for any time step size. Therefore, sufficiently high spatial resolution is indispensable in order to maintain stability, which increases the computational cost. To overcome these difficulties and develop a robust solver for high‐Reynolds‐number flow problem, it is necessary to further study the instability problem at small time steps. We numerically investigate the effect of the pressure gradient term in projection step and the velocity divergence term in pressure Poisson equation on the stability for small time step size, respectively, and conclude that the DG formulation of the pressure gradient term has a more significant effect on the stability of the scheme than that of the velocity divergence term. Integration by parts of these terms is essential in order to improve the stability of the scheme. Based on this discretization format, an appropriate penalty parameter for pressure Poisson equation is utilized so as to provide the scheme with an inf‐sup stabilization. Moreover, the lid‐driven cavity flow is considered to verify that this numerical algorithm enhances the stability without additional stabilization term at small time step size and high‐Reynolds number for equal‐order polynomial approximations.