Collaborative Research: Numerical Methods for Partial Differential Equations Arising in Shallow Water Modeling

本研究项目将为浅水计算方法及相关模型的发展做出重要贡献。将特别注意在海洋学、大气科学以及水力、海岸、土木和提高采油工程中出现的应用，其中必须考虑到底部地形的快速变化、科里奥利力、摩擦、非保守项和不确定现象。研究的问题包括城市地区的雨水排水和洪水、浊度流的浅水模型、多层流和数据不确定的浅水模型。正在开发的新工具有望在设计海岸保护系统和调查大陆架钻井平台上沉积物运输的影响方面具有巨大潜力，并有助于开发洪水缓解系统和规划新的城市地区。该项目旨在为浅水方程和相关模型开发准确、高效和稳健的数值方法，特别是涉及非光滑（不连续）解和涉及复杂非线性波、移动界面和不确定数据的问题。浅水模型是由依赖时间的偏微分方程（PDEs）组成的系统，这些偏微分方程是根据质量和动量守恒、流体静力或正压近似等物理性质推导出来的。当然，这些模型，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技术，例如在pde的原始系统中平衡的项之间的数值平衡（开发良好平衡的方案），确保所有流体层的正性（这对于准确描述干燥和近干燥状态以及执行非线性稳定性是绝对必要的），算子分裂方法，接口跟踪方法，以及其他将成为研究项目重点的方法。新技术的发展将基于高阶激波捕获有限体积方案、准确高效的ODE求解器和随机伽辽金方法，利用这些方法在研究问题的背景下的主要优势。

This research project will contribute significantly toward development of computational methods for shallow water and related models. Special attention will be paid to applications arising in oceanography, in atmospheric sciences, and in hydraulic, coastal, civil, and enhanced oil recovery engineering, in which rapid changes in the bottom topography, Coriolis forces, friction, nonconservative terms, and uncertain phenomena have to be taken into account. The problems under study include rainwater drainage and flooding in urban areas, shallow water models of turbidity currents, multilayer flows, and shallow water models with uncertain data. The new tools under development promise to have great potential in designing coastal protection systems and investigating the effects of sediment transport on shelf drilling platforms as well as contributing toward the development of flood mitigation systems and planning of new urban areas. The project is aimed at developing accurate, efficient, and robust numerical methods for shallow water equations and related models, with particular reference to problems that admit nonsmooth (discontinuous) solutions and involve complicated nonlinear waves, moving interfaces, and uncertain data. 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 models, especially in the cases of high space dimensions, require development and implementation of special numerical techniques such as numerical balancing between the terms that are balanced in the original system of PDEs (development of well-balanced schemes), ensuring positivity of all fluid layers (this is absolutely necessary for both accurate description of dry and near dry states and enforcement of nonlinear stability), operator splitting methods, interface tracking approaches, and others that will be in the focus of the research project. The development of new techniques will be based on high-order shock-capturing finite-volume schemes, accurate and efficient ODE solvers, and stochastic Galerkin methods, utilizing major advantages of each one of these methods in the context of the problems under study.