Low Mach preconditioned density-based methods with implicit Runge–Kutta schemes in physical-time

Abstract

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 accuracyorder 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.