Godunov-Type Central Schemes for Hyperbolic Problems: Further Development, Adaptation, and Applications

双曲问题的 Godunov 型中心方案：进一步发展、适应和应用

• 批准号: 0310585
• 负责人: Alexander Kurganov
• 金额: $ 13.07万
• 依托单位: Tulane University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0310585&HistoricalAwards=false
• 关键词: Godunov; Type; Central; Schemes; Hyperbolic

Central schemes may serve as universal finite-difference methods fornumerically solving hyperbolic conservation and balance laws,Hamilton-Jacobi equations, and related problems. Such schemes are nottied to the specific eigen-structure of the problem, and hence can beimplemented in a straightforward manner for a wide variety of nonlinearequations governing the spontaneous evolution of large gradientphenomena. This project aims to further improve the family of Godunov-type centralschemes, recently developed by Kurganov et al. The main ideas behind theconstruction of the new, less dissipative central schemes are to usemore precise estimate of the smooth and nonsmooth parts of the solutionby considering non-rectangular control volumes; to use a more accurateprojection of the evolved data onto the original, non-staggered grid;and to avoid the loss of information when very accurate fully-discreteschemes are reduced to a much simpler semi-discrete form.The second main goal of the project is the application of central schemesto various multi-phase and multi-fluid flow models, the Saint-Venantsystems of shallow water equations (which describe flows in rivers andcoastal areas), multi-layer shallow water systems, models of transport ofpollutant in shallow water, the Euler equations of gas dynamics subjectto a static gravitational field, chemotaxis models, reactive flows (inparticular, the models describing stiff detonation waves), extendedthermodynamics, shallow water equations on a rotating sphere, acousticwave propagation, heterogeneous elasticity, granular material flows.Naturally, these applications involve multiple space dimensions, complexgeometries and moving boundaries/interfaces, This would require furtherdevelopment of the theory and implementation of central schemes. Inparticular, semi-discrete central schemes on unstructured and triangularmeshes will be derived, and different adaptive techniques will beincorporated into the central framework.Recent development of modern technology requires reliable, efficient,high-resolution methods for solving time-dependent partial differential equations (PDEs), including multidimensional systems of hyperbolicconservation and balance laws, Hamilton-Jacobi equations, and relatedproblems. In the past decade, a family of simple, universal,Riemann-solver-free finite volume central schemes has proven to bean appealing alternative to the more complicated and problem orientedupwind schemes. The advantages of central schemes are particularlyprominent when they are used to solve complicated multidimensionalsystems of PDEs arising in such important fields including fluidmechanics, gas dynamics, geophysics, meteorology, magnetohydrodynamics,astrophysics, multi-component flows, granular flows, reactive flows,semiconductors, non-Newtonian flows, geometric optics, traffic flow,image processing, financial, biological modeling, differential games,and optimal control. This project is focused on the further development and improvement ofcentral schemes, and on their practical applications. The new centralschemes will be incorporated into a general-purpose adaptive meshrefinement (AMR) and adaptive moving mesh (AMM) codes, which will befreely accessible for the scientific and industrial communities. Thesecodes will serve as a reliable and robust "black-box-solver" for arather comprehensive class of time-dependent PDEs.

中央方案可以用作通用有限差异方法，可以求解双曲线保护和平衡法，汉密尔顿 - 雅各比方程和相关问题。此类方案与该问题的特定本征结构相关，因此可以直接地对大量的非线程进行大量的非线程，从而可以简单地进行大量的非线程。该项目旨在进一步改善Kurganov等人最近开发的Godunov型Centralschemes家族。新的，不太耗散的中央计划的建设背后的主要思想是考虑到非矩形控制量的解决方案的平滑和非平滑部分的精确估计；将进化的数据更准确地投入到原始的，非st依的网格上；并在将非常准确的完全限制化学降低到更简单的半污垢形式时避免信息丢失。该项目的第二个主要目标是中央图案的应用各种多相和多流体流模型的应用，浅水方程的圣化系统（描述了河流中的流动和沿岸地区），多层浅水系统，浅水运输模型气体动力学的EULER方程，以静态引力场，趋化模型，反应性流动（内部的动态，描述僵硬的爆炸波），延长的热动力学，旋转球体上的浅水方程，旋转的浅水等方程，声学之际传播，异构弹性，粒状物质流动，这些弹性。应用程序涉及多个空间维度，复合地理和移动边界/接口，这将需要进一步发展中心方案的理论和实施。将得出关于非结构化和三角形矩形的内部，半差异的中央计划，并将不同的自适应技术纳入中心框架中。现代技术的开发需要可靠，高效，高分辨率的方法来依赖时间依赖时间依赖分离的偏差（PDE）（PDES） ），包括多维的多维系统，平衡法，汉密尔顿 - 雅各比方程和相关问题。在过去的十年中，一个简单，普遍的，无摩尔 - 索的有限卷中央计划的家族已证明可以吸引更为复杂和面向问题的方案的替代品。当中央计划被用来解决在此类重要领域中引起的PDE的复杂多维系统时，其优势尤其重要 - 纽顿流，几何光学，交通流量，图像处理，财务，生物建模，差异游戏和最佳控制。该项目的重点是进一步的发展和改进的中心方案及其实际应用。新的CentralsChemes将纳入通用自适应网状（AMR）和自适应移动网格（AMM）代码中，这对于科学和工业社区来说将是可访问的。对于Arather综合时间依赖时间的PDE，这些序列将成为可靠且坚固的“黑盒溶剂”。