Efficient Hybridizable Discontinuous Galerkin Methods for Phase Field Fluid Models

多相流在自然现象和工业应用中普遍存在。常见的例子包括波浪破碎和晃动、污染物在含水层中的传输、石油工程中的石油回收、血液中的药物输送、燃烧反应器中的气固两相流动、聚合物电解质膜燃料电池技术中的废气管理等。扩散界面流体模型在多相流界面现象的数值模拟中得到了越来越广泛的应用。它们能够捕捉流体界面的平滑过渡，并且可以在固定网格上进行模拟，而不需要显式的界面跟踪。在求解扩散界面模型时，一个特殊的挑战是小宽度的扩散界面经常表现出气泡合并或分裂等不稳定性。传统的高阶方法容易在扩散界面附近产生虚假振荡，这会污染界面区域以外的数值解，甚至会由于负粘性、密度或迁移率而导致程序的爆破。本项目的目的是开发能够准确捕捉多相流运动界面的高阶数值方法。实际应用中既可以看到扩散主导的流动，也可以看到平流主导的流动。研究人员首先发展和分析了可证明超收敛的杂交间断Galerkin方法(HDG)，用于求解扩散为主区域中的扩散界面流体模型。设计的核心思想是用高次多项式逼近解变量，而不是数值迹和梯度变量，并探索基于局部投影的镇定。然后，PI设计了稳定的高阶HDG方法，该方法采用标量辅助变量(SAV)时间推进格式，用于平流为主的流动。该方法稳定了四阶非线性对流扩散方程中的平流，同时保持了基本的能量定律。稳定的SAV-HDG算法使扩散界面方法能够准确地捕捉平流为主区域中的尖锋和不稳定界面，并允许在每个时间步长高效地并行计算较小的系统。最后，PI开发并实现了扩散界面流体模型的快速非线性HDG多重网格求解器。实用解算器将进一步解决HDG方法缺乏高效迭代解算器/预条件的问题。研究生参与本项目的工作。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Multiphase flow is ubiquitous in natural phenomena and industrial applications. Common examples include wave-breaking and sloshing, contaminant transport in aquifers, oil recovery in petroleum engineering, drug delivery in blood flow, gas-particle flow in combustion reactors, exhaust management in Polymer Electrolyte Membrane fuel cell technology, and so forth. The diffuse interface fluid models have become increasingly popular in the numerical modeling of interfacial phenomena associated with multiphase flows. They are able to capture smooth transitions of fluid interface, and simulations can be carried out on a fixed grid without explicit interface tracking. A particular challenge in solving diffuse interface models is that the diffusive interface of small width often exhibits instability such as bubble merging or splitting. Traditional high order methods are prone to spurious oscillations around diffusive interfaces which can pollute the numerical solution beyond the interface region and even cause blow-up of the code due to negative viscosity, density or mobility. The aim of this project is to develop high order numerical methods that can accurately capture moving interfaces of multiphase flow.Real-world applications see both diffusion dominated flows and advection dominated flows. The investigator first develops and analyzes provably superconvergent hybridizable discontinuous Galerkin methods (HDG) for solving diffuse interface fluid models in the diffusion-dominated regime. The key idea in the design is to approximate solution variables by higher order polynomials than those for the numerical traces and gradient variables, and to explore local projection based stabilization. The PI then designs stabilized high order HDG methods effected with the Scalar Auxiliary Variable (SAV) time-stepping schemes for advection-dominated flows. The methods stabilize advection in the nonlinear fourth order advection-diffusion equation while preserve the underlying energy laws. The stabilized SAV-HDG algorithms enable diffuse interface methods to accurately capture sharp fronts and unstable interfaces in the advection-dominated regime, and allow efficient parallel computation of smaller systems at each time step. Finally the PI develops and implements fast nonlinear HDG multigrid solvers for diffuse interface fluid models. The practical solvers will further address the lack of efficient iterative solvers/preconditioners for HDG methods Graduate students participate in the work of this project.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.