As far as the authors are aware, low Mach preconditioned density-based methods found in the peer-reviewed literature only employ multi-step schemes for physical-time integration. This essentially limits the maximum achievable temporal accuracy-order of these methods to two, since the multi-step schemes of order higher than two are conditionally stable. However, the present paper shows how these methods can employ multi-stage schemes in physical-time using the same low Mach preconditioning techniques developed over the past few decades. In doing so, it opens up the rich field of Runge–Kutta time integration schemes to low Mach preconditioned density-based methods. One and two-dimensional test cases are used to demonstrate the capabilities of this novel approach. The former simulates the propagation of marginally stable entropy perturbations superposed on a uniform flow whereas the latter simulates the temporal growth of vorticity perturbations superposed on an absolutely unstable planar mixing-layer. A novel procedure is employed to generate highly accurate initial conditions for the two-dimensional test case, it minimizes receptivity regions as well as deleterious interactions with artificial boundary conditions due to the numerical error introduced by approximate initial conditions. These test cases show that second, third and fourth order multi-stage schemes with strong linear numerical stability can be successfully utilized for the physical-time integration of low Mach preconditioned density-based methods.

据作者所知，同行评审文献中发现的基于低马赫数预处理密度的方法仅采用多步方案进行物理时间整合。这基本上将这些方法的最大可实现时间精度阶数限制为两个，因为阶数高于两个的多步方案是条件稳定的。然而，本文展示了这些方法如何使用过去几十年开发的相同低马赫预处理技术在物理时间中采用多级方案。通过这样做，它为基于低马赫数预处理密度的方法开辟了 Runge-Kutta 时间积分方案的丰富领域。一维和二维测试用例用于演示这种新颖方法的功能。前者模拟叠加在均匀流上的边际稳定熵扰动的传播，而后者模拟叠加在绝对不稳定平面混合层上的涡度扰动的时间增长。一种新的程序被用来为二维测试用例生成高精度的初始条件，它最大限度地减少了由于近似初始条件引入的数值误差而导致的接收区域以及与人工边界条件的有害相互作用。这些测试案例表明，具有强线性数值稳定性的二阶、三阶和四阶多级方案可以成功地用于低马赫预处理密度方法的物理时间积分。