Consistent splitting schemes are among the most accurate pressure segregation methods, incurring no splitting errors or spurious boundary conditions. Nevertheless, their theoretical properties are not yet fully understood, especially when finite elements are used for the spatial discretisation. This work proposes a simple scalar auxiliary variable (SAV) technique that, when combined with standard finite elements in space, guarantees unconditional stability for first- and second-order consistent splitting schemes. The framework is implicit-explicit (IMEX) and only requires solving linear transport equations and a pressure Poisson problem per time step. Furthermore, pressure stability is attained with respect to a stronger norm than in classical projection schemes, which allows eliminating the inf-sup compatibility requirement on the velocity-pressure pairs. The accuracy of the new framework is assessed through numerical examples.

翻译：一致分裂格式是最精确的压力分离方法之一，不会引入分裂误差或虚假边界条件。然而，其理论性质尚未得到完全理解，特别是在使用有限元进行空间离散化时。本文提出了一种简单的标量辅助变量（SAV）技术，当与标准空间有限元结合时，能够保证一阶和二阶一致分裂格式的无条件稳定性。该框架采用隐式-显式（IMEX）方法，每个时间步仅需求解线性输运方程和一个压力泊松问题。此外，压力稳定性是在比经典投影格式更强的范数下实现的，从而消除了速度-压力对需满足inf-sup相容性条件的要求。通过数值算例验证了新框架的精度。