In 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.

中文翻译：

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一种基于高阶对偶分裂方案的稳定不连续伽辽金方法，不可压缩流动没有附加稳定项

在这项工作中，我们考虑基于高阶对偶分裂方案的不可压缩 Navier-Stokes 方程的等阶不连续 Galerkin (DG) 求解器。为了保持稳定，该方法的时间步长已被严格限制。时间步长的上限受 Courant–Friedrichs–Lewy (CFL) 条件的限制（Hesthaven 和 Warburton，2007），下限要求大于取决于雷诺数和空间分辨率的临界值（Ferrer et al ., 2014)。对于高雷诺数流动问题，如果空间分辨率低，临界值可能大于 CFL 条件，那么任何时间步长都会出现不稳定。因此，为了保持稳定性，足够高的空间分辨率是必不可少的，这增加了计算成本。为了克服这些困难并开发高雷诺数流动问题的鲁棒求解器，有必要进一步研究小时间步长的不稳定性问题。我们分别数值研究了投影步骤中的压力梯度项和压力泊松方程中的速度发散项对小时间步长稳定性的影响，并得出结论，压力梯度项的 DG 公式对该方案的稳定性高于速度发散项的稳定性。为了提高方案的稳定性，将这些项的部分集成是必不可少的。基于这种离散化格式，利用压力泊松方程的适当惩罚参数为方案提供 inf-sup 稳定性。而且，