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、高效和稳健的数值方法，特别是涉及非光滑（不连续）解的问题和涉及高度不同尺度的问题，因此难以用数值方法解决。浅水及其相关模式被广泛用作研究河流和沿海地区水流以及研究大气科学和海洋学中的各种现象的数学框架。这些模型是由依赖于时间的偏微分方程（PDEs）组成的系统，这些偏微分方程是利用诸如质量和动量守恒、流体静力或正压近似等物理性质推导出来的。拟研究的主要部分将集中于发展解决涉及复杂非线性波现象、复杂计算域和移动界面问题的新方法。由此产生的方法，虽然基于高阶激波捕获有限体积方案和非耗散无网格粒子方法，但将结合特殊的数值技术，例如在pde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都是绝对必要的）。准确高效的算子分割、准确高效的接口跟踪等将成为拟研究项目的重点。拟建项目将对浅水计算方法及相关模型的发展作出重大贡献。将特别注意在海洋学和大气科学中出现的应用，其中必须考虑到科里奥利力（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、海啸波的传播及其上岸，以及各种环境中污染物的传播。设计中的数值方法将为研究包括海啸和异常浪在内的各种内部和表面水波提供相当强大的工具。这些在深水和浅水中都出现的极端海浪对人类和基础设施的安全产生重大影响，并对船舶、石油平台、海岸线和海底造成破坏，并改变生物环境。因此，了解这些极端海浪的物理特性是一项重要的任务，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.