Collaborative Research: Structure Preserving Numerical Methods for Hyperbolic Balance Laws with Applications to Shallow Water and Atmospheric Models

该项目将为浅水和相关模型的计算方法的发展做出重大贡献，并将为研究各种水波和大气现象提供更强大的工具。将特别注意在海洋学，大气科学，水力，海岸，土木工程，其中在海底地形，科里奥利力，摩擦，多尺度制度，和不确定的现象的因素严重的快速变化所产生的应用。研究的问题将包括多连通河道系统中的浅水流动、海啸波传播和低弗劳德状态浅水模型，新开发的工具可能在设计海岸保护系统和调查沉积物输运对陆架钻井平台的影响方面具有很大的潜力，并有助于更好地预测热带气旋和云的动力学模型。该项目的重点是为双曲线平衡律开发新的结构保持数值方法，并将其应用于浅水方程和相关模型。浅水模式是一组时间相关的偏微分方程（PDE），这些偏微分方程是利用诸如质量和动量守恒、流体静力学或正压近似等物理性质推导出来的。自然，这些应用，特别是在高空间维度的情况下，需要开发和实施特殊的数值方法，不仅与偏微分方程的管理系统相一致，但也保持一定的结构和渐近性质的基本问题在离散水平。新的数值技术的发展将基于高阶激波捕捉有限体积法，渐近保持，自适应移动网格和随机Galerkin方法，利用这些方法中的每一个的主要优势，在研究的问题。除了提供证实数值方法的例子外，上述应用对当今科学中出现的广泛问题具有实质性的独立价值。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的知识价值和更广泛的影响审查标准进行评估来支持。

This project will significantly contribute toward development of computational methods for shallow water and related models and will provide considerably more powerful tools for studying a variety of water waves and atmospheric phenomena. Special attention will be paid to applications arising in oceanography, atmospheric sciences, hydraulic, coastal, civil engineering, in which rapid changes in the bottom topography, Coriolis forces, friction, multiscale regimes, and uncertain phenomena factor heavily. The studied problems will include shallow water flows in multi-connected river channel systems, tsunami wave propagation and low Froude regime shallow water models, dynamics models of tropical cyclones and clouds with uncertain data.The newly developed tools may have a great potential in designing coastal protection systems and investigating the effects of sediment transport on shelf drilling platforms as well as contributing to a better prediction of tropical cyclones trajectories and tsunami wave propagation and on-shore arrival.The project focuses on development of new structure preserving numerical methods for hyperbolic balance laws with applications to shallow water equations and related models. Shallow water models are systems of time-dependent partial differential equations (PDEs) that are derived using physical properties such as conservation of mass and momentum, and hydrostatic or barotropic approximations. Naturally, these applications, especially in cases of high space dimensions, require development and implementation of special numerical methods that are not only consistent with the governing system of PDEs, but also preserve certain structural and asymptotic properties of the underlying problem at the discrete level. The development of new numerical techniques will be based on high-order shock-capturing finite-volume schemes, asymptotic preserving, adaptive moving mesh and stochastic Galerkin methods utilizing major advantages of each one of these methods in the context of studied problems. Besides providing examples that corroborate the numerical approach, the foregoing applications are of a substantial independent value for a broad class of problems arising in today's science.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.