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

中心格式可以作为数值求解双曲守恒律和平衡律、Hamilton-Jacobi方程及相关问题的通用有限差分方法。这样的计划是nottied到特定的特征结构的问题，因此，可以被实施在一个简单的方式为各种各样的非线性方程的自发演变的大gradientphenomenon。本项目旨在进一步改进Kurganov等人最近发展的Godunov型中心格式族。构造新的、耗散较小的中心格式背后的主要思想是通过考虑非矩形控制体积来更精确地估计解的光滑和非光滑部分;该项目的第二个主要目标是将中心格式应用于各种多相和多流体流动模型，即浅水方程的圣维南系统（描述河流和海岸地区的流动），多层浅水系统，浅水污染物输运模型，静态重力场下的气体动力学欧拉方程，趋化性模型，反应流（特别是描述刚性爆轰波的模型）、扩展热力学、旋转球上的浅水方程、声波传播、非均匀弹性、颗粒材料流动等，这些应用自然涉及到多维空间、复杂几何和运动边界/界面，这就需要进一步发展理论和实现中心格式。特别是，半离散中心格式的非结构化和三角形网格，不同的自适应技术将被纳入到中心框架。现代技术的最新发展需要可靠的，有效的，高分辨率的方法来解决时间相关的偏微分方程（PDE），包括多维系统的双曲守恒和平衡律，Hamilton-Jacobi方程，以及相关的问题。在过去的十年中，一族简单的，通用的，Riemann求解器的自由有限体积中心格式已被证明是一个有吸引力的替代更复杂的和面向问题的逆风格式。中心格式的优点是特别突出的，当他们被用来解决复杂的多维系统的偏微分方程出现在这样的重要领域，包括流体力学，气体动力学，物理学，气象学，磁流体力学，天体物理学，多组分流，颗粒流，反应流，半导体，非牛顿流，几何光学，交通流，图像处理，金融，生物建模，微分对策和最优控制。本项目的重点是中心格式的进一步发展和改进，以及它们的实际应用。新的中心方案将被纳入一个通用的自适应网格细化（AMR）和自适应移动网格（AMM）代码，这将是免费提供给科学和工业界。这些代码将作为一个可靠的和强大的“黑箱求解器”的arather全面类的时间依赖偏微分方程。

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.