Efficient Hybridizable Discontinuous Galerkin Methods for Phase Field Fluid Models

多相流在自然现象和工业应用中无处不在。常见的例子包括破波和晃动、含水层中的污染物输送、石油工程中的采油、血流中的药物输送、燃烧反应器中的气粒流、聚合物电解质膜燃料电池技术中的废气管理等等。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。求解扩散界面模型的一个特殊挑战是，小宽度的扩散界面往往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周围产生伪振荡，从而污染界面区域以外的数值解，甚至由于负粘度、负密度或负迁移率导致代码爆炸。本项目的目的是发展能够准确捕捉多相流运动界面的高阶数值方法。在实际应用中，扩散流和平流流都占主导地位。研究人员首先开发并分析了可证明的超收敛杂交不连续伽辽金方法（HDG），用于求解扩散主导状态下的扩散界面流体模型。设计的核心思想是用比数值轨迹和梯度变量更高阶的多项式逼近解变量，并探索基于局部投影的稳定。在此基础上，针对平流占主导地位的流动，设计了受标量辅助变量（SAV）时步格式影响的稳定高阶HDG方法。该方法稳定了非线性四阶平流扩散方程中的平流，同时保持了底层能量定律。稳定的SAV-HDG算法使漫射界面方法能够准确捕获平流主导状态下的尖锐锋面和不稳定界面，并允许在每个时间步长对较小系统进行有效的并行计算。最后，PI开发并实现了用于扩散界面流体模型的快速非线性HDG多网格求解器。实际解算器将进一步解决HDG方法缺乏有效的迭代解算器/预处理器的问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.