We develop a multirate timestepper for semi-implicit solutions of the unsteady incompressible Navier-Stokes equations (INSE) based on a recently-developed multidomain spectral element method (SEM). For {\em incompressible} flows, multirate timestepping (MTS) is particularly challenging because of the tight coupling implied by the incompressibility constraint, which manifests as an elliptic subproblem for the pressure at each timestep. The novelty of our approach stems from the development of a stable overlapping Schwarz method applied directly to the Navier-Stokes equations, rather than to the convective, viscous, and pressure substeps that are at the heart of most INSE solvers. Our MTS approach is based on a predictor-corrector (PC) strategy that preserves the temporal convergence of the underlying semi-implicit timestepper. We present numerical results demonstrating that this approach scales to an arbitrary number of overlapping grids, accurately models complex turbulent flow phenomenon, and improves computational efficiency in comparison to singlerate timestepping-based calculations.
翻译:我们为基于最近开发的多域光谱元素法(SEM)的不固定的纳维埃-斯托克方程式(INSE)开发了一个半不固定的半不透明溶解的多时间步数。对于挤压流,多度时间步数(MTS)特别具有挑战性,因为压抑性制约意味着紧密结合,这表现为每个时间步数的压力的椭圆次问题。我们的方法的新颖性源于发展一种稳定重叠的施瓦兹法,直接适用于纳维-斯托克方程式,而不是适用于大多数INSE解决者核心的对流、对流和压力子步数。我们的MTS方法基于一种预测-纠正器(PC)战略,它保持了基础半隐含时间步数的暂时结合。我们提出了数字结果,表明这一方法的规模是任意的重叠格数,精确的模型复杂的动荡流现象,并且提高了计算效率,以单级制时间步计算为基础计算为基础。