Abstract A modified version of the projection scheme [19] is proposed, which does not show a lower limit for the time step in contrast to the limits of stability observed numerically for some projection type schemes. An advantage of the proposed scheme is that the right-hand side of the Poisson equation for the pressure is independent of the time step. An explicit version of the current scheme is also provided besides the implicit-explicit one. For the implicit-explicit version, we retain divergence of the viscous terms on the right-hand side of the Poisson equation in order to achieve a higher accuracy for low Reynolds number flows. In this way, we also ensure that the Poisson equation with Neumann boundary condition is consistent on the discrete level, where we discretize the boundary condition as well. The spatial discretization is performed for equal and mixed orders of the velocity and pressure using the dG method. Long-term stability and optimal temporal and spatial convergence rates are obtained for the unsteady Taylor vortex and periodic channel flows. An excellent agreement with the similarity solution is achieved for the steady plane stagnation point flow. Finally, the current projection scheme is compared with the original one for a more complex coupled electro-fluid-dynamics problem. For the same numerical settings, the current scheme appears to be more accurate in predicting the expected physical behavior and it is long-term stable while the original scheme fails.

中文翻译：

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使用高阶 dG 方法的非定常不可压缩 Navier-Stokes 方程的隐显和显式投影方案

摘要 提出了投影方案 [19] 的修改版本，与某些投影类型方案在数值上观察到的稳定性极限相比，它没有显示时间步长的下限。所提出方案的一个优点是压力泊松方程的右侧与时间步长无关。除了隐式-显式之外，还提供了当前方案的显式版本。对于隐式-显式版本，我们保留泊松方程右侧粘性项的发散，以便在低雷诺数流中获得更高的精度。通过这种方式，我们还确保了具有 Neumann 边界条件的 Poisson 方程在离散水平上是一致的，我们也对边界条件进行了离散化。使用 dG 方法对速度和压力的等阶和混合阶进行空间离散化。对于不稳定的泰勒涡流和周期性通道流，获得了长期稳定性和最佳时间和空间收敛率。对于稳态平面驻点流，实现了与相似解的极好一致性。最后，针对更复杂的耦合电流-流体动力学问题，将当前的投影方案与原始方案进行比较。对于相同的数值设置，当前方案在预测预期物理行为方面似乎更准确，并且在原始方案失败的情况下长期稳定。对于不稳定的泰勒涡流和周期性通道流，获得了长期稳定性和最佳时间和空间收敛率。对于稳态平面驻点流，实现了与相似解的极好一致性。最后，针对更复杂的耦合电流-流体动力学问题，将当前的投影方案与原始方案进行比较。对于相同的数值设置，当前方案在预测预期物理行为方面似乎更准确，并且在原始方案失败的情况下长期稳定。对于不稳定的泰勒涡流和周期性通道流，获得了长期稳定性和最佳时间和空间收敛率。对于稳态平面驻点流，实现了与相似解的极好一致性。最后，针对更复杂的耦合电流-流体动力学问题，将当前的投影方案与原始方案进行比较。对于相同的数值设置，当前方案在预测预期物理行为方面似乎更准确，并且在原始方案失败的情况下长期稳定。