High Order Maximum Principle Preserving Finite Difference Schemes for Hyperbolic Conservation Laws

高阶极大值原理保持双曲守恒定律的有限差分格式

• 批准号: 1316662
• 负责人: Zhengfu Xu
• 金额: $ 22.63万
• 依托单位: Michigan Technological University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-09-01 至 2017-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1316662&HistoricalAwards=false
• 关键词: Order; Maximum; Principle; Preserving; Finite

The main focus of this proposal is to develop and to analyze a novel parametrized maximum principle preserving flux limiter technique for high order numerical schemes applied to hyperbolic conservation laws. A flux limiting technique will also be designed to obtain high order positivity preserving schemes. Numerical schemes that preserve the maximum principle and positivity are desirable because physically relevant solutions have those properties. The development is based on finite difference methods, which have the advantage of producing accurate approximations with low computational cost especially in multi-dimensional simulations. Within the proposed framework, conservative maximum principle preserving high order finite difference, finite volume and discontinuous Galerkin schemes can be designed that allow for significantly large CFL number, and therefore more efficient computational simulation. Some important applications investigated in this proposal include compressible Euler equations, magneto hydrodynamics equations and Vlasov-Maxwell equations.The investigator is developing new computational techniques that can be applied to difficult and very important problems in science and engineering. These techniques address shortcomings in existing methods and should allow more efficient, robust, and accurate computer simulations in a number of critical applications. One such application is the supersonic flow problem, which is of great importance in designing astrophysical jets and also in the simulation of reentry vehicle for space flight modeling. Another application is the study of the magneto hydrodynamic systems, which arises in space weather modeling, in modeling electric propulsion sources, and in systems involving plasma (such as plasma-opening switches, flight control via plasma, plasma assisted combustion). These problems are strategically important for the design of the next generation devices of great industrial and commercial value.

该提案的主要重点是开发和分析一种新型的参数性最大原理，以保留用于双曲线保护定律的高阶数值方案的通量限制器技术。通量限制技术也将设计为获得高阶阳性保留方案。保留最大原理和积极性的数值方案是可取的，因为与物理相关的解决方案具有这些特性。该开发基于有限差异方法，该方法具有产生较低计算成本的准确近似值的优点，尤其是在多维模拟中。在拟议的框架内，可以设计保守的最大原理，以保留高阶差异，有限的体积和不连续的Galerkin方案，可以设计出明显较大的CFL数量，因此可以更有效地进行计算模拟。该提案中研究的一些重要应用包括可压缩的Euler方程，磁铁流体动力学方程和Vlasov-Maxwell方程。研究人员正在开发新的计算技术，这些技术可以应用于科学和工程中的困难和非常重要的问题。这些技术解决了现有方法中的缺点，应允许在许多关键应用程序中更有效，稳健和准确的计算机模拟。一个这样的应用是超音速流问题，这对于设计天体物理喷气机以及用于太空飞行建模的重新进入车辆非常重要。另一个应用是对磁铁流体动力系统的研究，该系统是在太空天气建模，对电推进源建模以及涉及等离子体的系统（例如等离子体开关，通过等离子体的飞行控制，等离子体辅助燃烧）的研究。这些问题在战略上对于伟大的工业和商业价值的下一代设备的设计在战略上很重要。