Collaborative Research: Numerical methods for Shallow Water Equations and Related Models

该项目旨在为浅水方程和相关模型开发准确，高效和鲁棒的数值方法，特别是涉及非光滑（不连续）解决方案的问题以及涉及高度不同尺度的问题，因此难以数值求解。浅水和相关模式被广泛用作研究河流和沿海地区水流以及大气科学和海洋学中各种现象的数学框架。这些模型是使用质量和动量守恒以及流体静力学或正压近似等物理性质推导出的与时间相关的偏微分方程（PDE）系统。拟议的研究的主要部分将集中在解决问题的新方法的发展，涉及复杂的非线性波现象，复杂的计算域和移动接口的问题。由此产生的方法，虽然基于高阶激波捕捉有限体积格式和非耗散无网格粒子方法，将纳入特殊的数值技术，如数值平衡之间的平衡，在原来的系统的偏微分方程（制定平衡的计划），确保所有流体层的正性（这对于精确描述干态和近干态以及加强非线性稳定性都是绝对必要的），精确和有效的算子分裂，准确和有效的界面跟踪，以及其他将成为拟议研究项目重点的问题。拟议的项目将大大有助于发展浅水和相关模型的计算方法。将特别注意海洋学和大气科学中出现的应用，其中必须考虑科里奥利力（由于地球自转），热力学和湍流效应。正在研究的问题除其他外包括大气锋和洋流的形成和传播、海啸波的传播及其抵达海岸以及污染物在各种环境中的传播。设计中的数值方法将为研究各种内波和表面波（包括海啸和流氓波）提供更强大的工具。这些在深水和浅水中出现的极端波浪对人员和基础设施的安全产生重大影响，并对船舶、石油平台、海岸线和海底造成损害，并改变生物环境。因此，了解这些极端波的物理学是一项重要的任务，甚至可能有助于拯救生命。

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 non-smooth (discontinuous) solutions and to problems that involve highly disparate scales and therefore are difficult to solve numerically. Shallow water and related models are widely used as a mathematical framework to study water flows in rivers and coastal areas as well as to investigate a variety of phenomena in atmospheric sciences and oceanography. These 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. The principal part of the proposed research will be focused on the development of new methods for solving problems involving complicated nonlinear wave phenomena, problems with complex computational domains and moving interfaces. The resulting methods, while based on high-order shock-capturing finite-volume schemes and non-dissipative mesh-free particle methods, will incorporate 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), accurate and efficient operator splitting, accurate and efficient interface tracking, and others that will be in the focus of the proposed research project. The proposed 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 and atmospheric sciences, in which the Coriolis forces (due to the Earth's rotation), thermodynamics, and turbulent effects have to be taken into account. The problems under study include, among others, formation and propagation of atmospheric fronts and ocean currents, propagation of tsunami waves and their on-shore arrival, as well as propagation of pollutants in various environments. The numerical methods under design will provide considerably more powerful tools for studying a variety of internal and surface water waves, including tsunami and rogue waves. These extreme waves, which arise both in deep and shallow water, have a significant impact on the safety of people and infrastructure, and are responsible for damage to ships, oil platforms, coastlines, and sea bottoms and for changes to the biological environment. Thus, understanding the physics of these extreme waves is an important task that may even contribute to saving lives.