Collaborative Research: High Order Numerical Schemes for Multi-Dimensional Systems of Conservation Laws and for Simulations of Multi-Phase Fluids

合作研究：守恒定律多维系统和多相流体模拟的高阶数值方案

• 批准号: 0107419
• 负责人: Xu-Dong Liu
• 金额: $ 12.91万
• 依托单位: University of California-Santa Barbara
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-08-15 至 2004-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0107419&HistoricalAwards=false
• 关键词: Collaborative; Research; Order; Numerical; Schemes

The main theme of the proposed project is the construction of high order accurate numerical schemes for solving multi-dimensional hyperbolic systems of conservation laws, and in particular the construction of numerical schemes for simulations of multi-phase fluid flows. This includes numerical methods for compressible flow, incompressible flow and heat transfer. Recently, the PI's introduced a boundary condition capturing method for variable coefficient Poisson equation in the presence of interfaces. The method is implemented using a standard finite difference discretization on a Cartesian grid making it simple to apply in several spatial dimensions. Furthermore, the resulting linear system is symmetric positive definite allowing for straightforward application of standard "black box" solvers, for example, multi-grid methods. Most importantly, this new method does not suffer from the numerical smearing. Using this method, the PI's extended the Ghost Fluid Method to treat two-phase incompressible flows, in particular those consisting of water and air. The numerical experiments show that the new numerical method performs quite well in both two and three spatial dimensions. Currently, they are working on extending this method to treat a wide range of problems, including for example combustion. Of particular interest is the extension of this method to include interface motion governed by the Cahn-Hilliard equation which models the non-zero thickness interface with a molecular force balance model. This proposed research on computational fluid dynamics is focused on the design, implementation and testing of new methods for simulating fluids such as water and gas using the computer. In particular, this work addresses problems where more than one type of one phase of fluid exist, e.g. mixtures of water and air. Our interest lies in improving the current state of the art algorithms so that they are better able to treat the interface that separates two fluids such as oil and water. The results of this research should be of interest to both the military (e.g. many naval applications involve the study of water and air mixtures) and to private industry. A particularly interesting example involves the interaction of water and oil in an underground oil recovery process. The research covered in this proposal has implications for math and science education as well. Not only will the PI's be working with and training graduate students in applied mathematics and engineering, but their research in extending these techniques to other fields, such as computer graphics, can play a role attracting the next generation of young scientists. For example, figure 7 in "Foster and Fedkiw, Practical Animation of Liquids, SIGGRAPH 2001" shows the lovable character "Shrek", from the feature film of the same name, taking a bath in mud.

拟议项目的主题是建设高精度的数值方案求解多维双曲守恒律系统，特别是建设的数值方案模拟多相流体流动。 这包括可压缩流动、不可压缩流动和传热的数值方法。 最近，PI的介绍了一种边界条件捕获方法的变系数泊松方程中存在的接口。 该方法是使用一个标准的有限差分离散化的笛卡尔网格，使其简单地应用在几个空间维度。 此外，所得到的线性系统是对称正定的，允许直接应用标准的“黑箱”求解器，例如，多重网格方法。 最重要的是，这种新方法不会受到数值拖尾的影响。 使用这种方法，PI的扩展幽灵流体方法来处理两相不可压缩流，特别是那些由水和空气组成的。 数值实验表明，新的数值方法在二维和三维空间都有很好的表现。 目前，他们正在努力将这种方法扩展到处理各种问题，包括燃烧。 特别令人感兴趣的是，这种方法的扩展，包括界面运动的Cahn-Hilliard方程的模型的非零厚度界面与分子力平衡模型。计算流体动力学的研究重点是设计，实施和测试使用计算机模拟水和气体等流体的新方法。 特别地，这项工作解决了存在多于一种类型的一相流体的问题，例如水和空气的混合物。 我们的兴趣在于改进当前最先进的算法，使它们能够更好地处理分离两种流体（如油和水）的界面。 这项研究的结果应该是感兴趣的军事（例如，许多海军应用涉及水和空气混合物的研究）和私营企业。 一个特别有趣的例子涉及地下采油过程中水和油的相互作用。 这项提案中所涵盖的研究也对数学和科学教育产生了影响。 PI不仅将与应用数学和工程的研究生合作并对其进行培训，而且他们将这些技术扩展到其他领域（如计算机图形学）的研究可以吸引下一代年轻科学家。 例如，在“Foster and Fedkiw，Practical Animation of Liquids，SIGGRAPH 2001”中的图7显示了来自同名故事片的可爱角色“史莱克”在泥中洗澡。