Multiphase flows are an important class of fluid flow and their study facilitates the development of diverse applications in industrial, natural and biomedical systems. Simulating such flows requires significant computational resources, making it prudent to devise an adaptive mesh refinement (AMR) method to mitigate this burden. We use a mathematical model that takes a continuum mechanical approach to describe multiphase mixture flows. The resulting system of equations poses numerical challenges due to the presence of multiple non-linear terms and a co-incompressibility condition, while the resulting fluid dynamics necessitate the development of an adaptive mesh refinement technique to accurately capture regions of interest while keeping computational costs low. We present an accurate, robust, and efficient computational method for simulating multiphase mixtures on adaptive grids, and utilize a multigrid solver to precondition the saddle-point system. We demonstrate that the AMR solver asymptotically approaches second order accuracy in $L^1$, $L^2$ and $L^\infty$ norms for all solution variables of the Newtonian and non-Newtonian models. All experiments demonstrate the solver is stable provided the time step size satisfies the imposed CFL condition. The solver can accurately resolve sharp gradients in the solution and, with the multigrid preconditioner, the solver behavior is independent of grid spacing. Our AMR solver offers a major cost savings benefit, providing up to 10x speedup in the numerical experiments presented here, with greater speedup possible depending on the problem set-up.

翻译：多相流是一类重要的流体流动，其研究有助于推动工业、自然和生物医学系统中多样化应用的发展。模拟此类流动需要大量计算资源，因此设计自适应网格细化（AMR）方法来减轻这一负担是审慎之举。我们采用基于连续介质力学方法的数学模型来描述多相混合物流动。所得方程组因存在多个非线性项和共不可压缩条件而带来数值挑战，同时所产生的流体动力学特性需要开发自适应网格细化技术，以在保持计算成本较低的情况下精确捕获感兴趣区域。我们提出了一种在自适应网格上模拟多相混合物的精确、鲁棒且高效的计算方法，并利用多重网格求解器对鞍点系统进行预处理。我们证明，对于牛顿和非牛顿模型的所有解变量，AMR求解器在$L^1$、$L^2$和$L^\infty$范数下渐近地达到二阶精度。所有实验表明，只要时间步长满足所施加的CFL条件，求解器就能保持稳定。该求解器能够精确解析解中的陡峭梯度，并且借助多重网格预处理器，其行为与网格间距无关。我们的AMR求解器具有显著的成本节约优势，在本文展示的数值实验中可实现高达10倍的加速，且根据问题设置的不同还可能获得更高的加速比。