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

DMS-9805546刘旭东该项目涉及几种求解多维守恒律系统的新数值方法，包括用于多相流体计算的新的完全保守方案。该项目的第一个贡献是将Friedrich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方针之一。构造了一族正格式。正格式是非常健壮、简单和低成本的。许多数值实验表明，正格式是最好的二阶精度高分辨率方法之一。这也是第一次工作，这种类型的理论结果，其中包含计划设计多维双曲系统。该项目的第二个贡献是多维双曲型方程组的新的凸非振荡（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）不需要完备的特征向量集，因此可以求解弱双曲方程组。(2)在每个空间维度上，逐场限制的速度是逐场限制的两倍，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。（4）凸ENO格式具有很强的鲁棒性。该项目的第三个贡献是引入了一种用于多相流问题的完全保守方法。这个新的想法，使我们能够避免寄生振荡附近的材料界面共同的所有其他保守计划。这是通过增加一个通用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，如伽马律气体，水和JWL（爆炸性材料）。新的思想适用于任何空间维度，并且是模式无关的，这意味着它应该适用于典型的用户现有代码。初步的数值实验表明，该格式是很有前途的。该项目旨在解决真实的世界问题，并打算对半导体器件建模、水下和固体爆炸物建模、计算流体力学、磁流体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获得更高的效率、简单性和鲁棒性;（2）改进多相流体的计算机模拟。本项目中报告的方法是朝着这一目标迈出的一步。

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.