High Order Number Schemes for Multi-Dimensional Systems of Conservation Laws and Conservative Schemes for MultiphaseFluids

多维守恒定律系统的高阶数方案和多相流体的保守方案

• 批准号: 9805546
• 负责人: Xu-Dong Liu
• 金额: --
• 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: 1998
• 资助国家: 美国
• 起止时间: 1998-07-01 至 2001-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9805546&HistoricalAwards=false
• 关键词: Order; Number; Schemes; Multi; Dimensional

DMS-9805546 Xu-Dong Liu This project is concerned with several new numerical methods for solving multidimensional systems of conservation laws, including a new fully conservative scheme for multi-phase fluid calculations. The first contribution of this project is the extension of Friedrich's positivity principle from multi-dimensional symmetric linear systems to systems of conservation laws, which has become one of the guidelines for designing numerical methods. A family of positive schemes is constructed. Positive schemes are very robust, simple and of low cost. Many numerical experiments have shown that positive schemes are among the best 2nd order accurate high resolution methods. This is also the first work of this type which contains theoretical results for scheme design in multi-dimensional hyperbolic systems. The second contribution of this project is the new Convex Essential-Non-Oscillatory (CENO) schemes for multi-dimensional hyperbolic systems. The scheme can be implemented in component-wise fashion. Therefore its apparent advantages are: (1) No complete set of eigenvectors is needed and hence weakly hyperbolic systems can be solved. (2) Component-wise limiting is twice as fast as field-by-field limiting in each space dimension, which makes Convex ENO one of the fastest existing schemes. (3) The component-wise version of the scheme is simple to program. In addition, (4) the Convex ENO scheme is very robust. The third contribution of this project is the introduction of a fully conservative method for multi-phase flow problems. This new idea enables us to avoid the spurious oscillations near material interfaces common to all other conservative schemes. This is done through the addition of a general equation of state. The new scheme works essentially for mixture of any fluids such as gamma-law gas, water and JWL (explosive material). The new idea works in any space dimension and is scheme-independent, which means it should apply to a typical users' existi ng code. Preliminary numerical experiments show that this scheme is very promising. This project is aimed at solving real world problems and is intended to have a significant impact on semiconductor device modeling, underwater and solid explosives modeling, computational fluid dynamics, magneto-hydrodynamics, and many other applications, which are all a part of high-performance computing. The principal goals are: (1) to design and improve numerical methods for more efficiency, simplicity, and robustness; (2) to improve computer simulation of multi-phase fluids. The methods reported in this project are a step towards this goal.

本项目研究了求解守恒律多维系统的几种新的数值方法，包括多相流体计算的一种新的全保守格式。本项目的第一个贡献是将弗里德里希的正性原理从多维对称线性系统扩展到守恒律系统，这已成为设计数值方法的指导原则之一。构造了一类正方案。正方案是非常强大的，简单和低成本。许多数值实验表明，正格式是最好的二阶精确高分辨率方法之一。这也是第一个包含多维双曲系统方案设计理论结果的此类工作。本课题的第二个贡献是新的多维双曲系统的凸本质非振荡（CENO）格式。该方案可以以组件方式实现。因此，它的明显优点是：(1)不需要特征向量的完备集，因此可以求解弱双曲系统。(2)在每个空间维度上，逐字段限制的速度是逐字段限制的两倍，这使得凸ENO成为现有最快的方案之一。(3)该方案的组件版本易于编程。此外，(4)凸ENO方案具有很强的鲁棒性。本项目的第三个贡献是引入了多相流问题的全保守方法。这种新思想使我们能够避免所有其他保守方案在材料界面附近常见的伪振荡。这是通过加入一般的状态方程来实现的。新方案基本上适用于任何流体的混合物，如伽马定律气体、水和JWL（爆炸性物质）。这个新想法适用于任何空间维度，并且与方案无关，这意味着它应该适用于典型用户的现有代码。初步的数值实验表明，该方案是很有前途的。该项目旨在解决现实世界的问题，旨在对半导体器件建模、水下和固体炸药建模、计算流体动力学、磁流体动力学和许多其他应用产生重大影响，这些都是高性能计算的一部分。主要目标是：(1)设计和改进数值方法，以提高效率、简单性和鲁棒性；(2)改进多相流体的计算机模拟。在这个项目中报告的方法是朝着这个目标迈出的一步。