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: 0106694
• 负责人: Ron Fedkiw
• 金额: $ 7.57万
• 依托单位: Stanford University
• 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=0106694&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.

拟议项目的主要主题是构建用于求解多维双曲型守恒律组的高精度数值格式，特别是构建用于模拟多相流体流动的数值格式。这包括可压缩流动、不可压缩流动和传热学的数值方法。最近，PI提出了一种具有界面的变系数泊松方程的边界条件捕捉方法。该方法是在笛卡尔网格上使用标准的有限差分离散实现的，这使得它在几个空间维度上的应用变得简单。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，例如多重网格法。最重要的是，这种新方法不会受到数值抹黑的影响。利用这种方法，PI将Ghost流体方法推广到处理两相不可压缩流动，特别是水和空气组成的两相流动。数值实验表明，新的数值方法在二维和三维空间上都有很好的效果。目前，他们正致力于将这种方法扩展到处理广泛的问题，例如燃烧。特别令人感兴趣的是这种方法的扩展，包括由Cahn-Hilliard方程控制的界面运动，该方程用分子力平衡模型模拟非零厚度界面。计算流体力学的研究重点是利用计算机模拟水和气体等流体的新方法的设计、实现和测试。特别是，这项工作解决了存在一种以上类型的一种流体的问题，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，以便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会引起军方(例如，许多海军应用涉及水和空气混合物的研究)和私营企业的兴趣。一个特别有趣的例子涉及地下石油开采过程中水和油的相互作用。这项提案中涉及的研究对数学和科学教育也有影响。PI不仅将与应用数学和工程方面的研究生合作并进行培训，而且他们将这些技术扩展到其他领域的研究，如计算机图形学，可以发挥吸引下一代年轻科学家的作用。例如，《Foster and Fedkiw，Practical Animation of Liquid，SIGGRAPH 2001》中的图7显示了同名故事片中可爱的角色史莱克在泥巴中洗澡。