具有电边界条件的 Navier-Stokes 方程压力 Poisson 方程重构的高阶方法
High-order Methods for a Pressure Poisson Equation Reformulation of the Navier-Stokes Equations with Electric Boundary Conditions
AI总结：
本文针对带 EBC 的 NSE 的 PPE 重构，结合 IMEX 时间步进与混合有限元，实现压力求解和速度更新解耦，并验证其在时空上至少达到三阶精度。
摘要：
不可压缩 Navier-Stokes 方程（NSE）的压力 Poisson 方程（PPE）重构通过关于压力的 Poisson 方程以及适当选择的边界条件来替代不可压缩性约束。这产生了一个仅关于速度场的时间演化方程，其中压力梯度充当非局部算子。因此，原则上，基于 PPE 重构的数值方法在实现高阶精度方面没有限制。本文研究了在多大程度上可以从具有电边界条件（EBC）的特定 PPE 重构中获得 NSE 的高阶方法。为此，采用隐式-显式（IMEX）时间步进将压力求解与速度更新解耦，同时避免抛物型时间步长限制；并在空间上使用混合有限元，以捕捉 EBC 所施加的结构。通过数值算例表明，该方法在空间和时间上均可产生至少三阶精度。
英文摘要：
Pressure Poisson equation (PPE) reformulations of the incompressible Navier-Stokes equations (NSE) replace the incompressibility constraint by a Poisson equation for the pressure and a suitable choice of boundary conditions. This yields a time-evolution equation for the velocity field only, with the pressure gradient acting as a nonlocal operator. Thus, numerical methods based on PPE reformulations, in principle, have no limitations in achieving high order. In this paper, it is studied to what extent high-order methods for the NSE can be obtained from a specific PPE reformulation with electric boundary conditions (EBC). To that end, implicit-explicit (IMEX) time-stepping is used to decouple the pressure solve from the velocity update, while avoiding a parabolic time-step restriction; and mixed finite elements are used in space, to capture the structure imposed by the EBC. Via numerical examples, it is demonstrated that the methodology can yield at least third order accuracy in space and time.
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