The pressure-correction method is a well established approach for simulating unsteady, incompressible fluids. It is well-known that implicit discretization of the time derivative in the momentum equation e.g. using a backward differentiation formula with explicit handling of the nonlinear term results in a conditionally stable method. In certain scenarios, employing explicit time integration in the momentum equation can be advantageous, as it avoids the need to solve for a system matrix involving each differential operator. Additionally, we will demonstrate that the fully discrete method can be expressed in the form of simple matrix-vector multiplications allowing for efficient implementation on modern and highly parallel acceleration hardware. Despite being a common practice in various commercial codes, there is currently no available literature on error analysis for this scenario. In this work, we conduct a theoretical analysis of both implicit and two explicit variants of the pressure-correction method in a fully discrete setting. We demonstrate to which extend the presented implicit and explicit methods exhibit conditional stability. Furthermore, we establish a Courant-Friedrichs-Lewy (CFL) type condition for the explicit scheme and show that the explicit variant demonstrate the same asymptotic behavior as the implicit variant when the CFL condition is satisfied.

翻译：压力校正方法是模拟非定常不可压缩流体的成熟方法。众所周知，对动量方程中的时间导数进行隐式离散化（例如使用后向差分公式并显式处理非线性项）会得到条件稳定方法。在某些情况下，在动量方程中采用显式时间积分具有优势，因为它避免了求解包含每个微分算子的系统矩阵。此外，我们将证明全离散方法可以表示为简单的矩阵-向量乘法形式，从而能够在现代高度并行的加速硬件上高效实现。尽管在各种商业代码中这是常见做法，但目前尚无关于此场景误差分析的文献。在本工作中，我们对压力校正方法的隐式及两种显式变体在全离散框架下进行了理论分析。我们论证了所提出的隐式和显式方法在何种程度上表现出条件稳定性。此外，我们为显式格式建立了柯朗-弗里德里希斯-列维（CFL）型条件，并证明当CFL条件满足时，显式变体具有与隐式变体相同的渐近行为。