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型守恒律和平衡律、Hamilton-Jacobi方程及相关问题的通用有限差分方法。这类方案不依赖于问题的特定本征结构，因此可以直接用于控制大梯度现象自发演化的各种非线性响应。该项目旨在进一步改进最近由Kurganov等人开发的Godunov型中心方案家族。构造新的、耗散较少的中心格式的主要思想是，通过考虑非矩形控制体，对解的光滑和非光滑部分进行更精确的估计；将演变的数据更准确地投影到原始的、非交错的网格上；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、圣文氏浅水方程组(描述河流和沿海地区的流动)、多层浅水系统、污染物在浅水中的传输模型、静重力场作用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、旋转球面上的浅水方程、声波传播、非均质弹性、物质流动。这些应用涉及多个空间维度、复杂的几何形状和移动的边界/界面，这将需要进一步发展中心方案的理论和实施。特别是，非结构网格和三角网格上的半离散中心格式将被推导出来，不同的自适应技术将被结合到中心框架中。现代技术的发展要求可靠、高效、高分辨率的方法来求解含时偏微分方程组，包括双曲型守恒律和平衡律的多维系统、哈密顿-雅可比方程和相关问题。在过去的十年中，一族简单的、通用的、无Riemann解算器的有限体积中心格式被证明是比更复杂和面向问题的迎风格式更有吸引力的替代格式。当中心格式用于求解在流体力学、气体动力学、地球物理、气象学、磁流体力学、天体物理、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等重要领域中出现的复杂的多维偏微分方程组时，其优势尤为突出。本项目的重点是中央方案的进一步发展和完善，以及它们的实际应用。新的中央方案将被纳入通用的自适应网格加密(AMR)和自适应移动网格(AMM)代码，科学界和工业界将可以免费访问这些代码。这些代码将作为一种可靠而健壮的“黑箱解算器”，用于求解一类全面的依赖时间的偏微分方程组。

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.