Adaptive multinumeric finite element methods for shallow water flow

沿海海洋建模的最新进展强调了两个主要主题：1)使用相对较大的计算域，这些计算域包含比特定兴趣区域大得多的区域，主要概念是将开放海洋边界放置在远离深水的非共振海洋盆地中；2)策略性地使用非结构化网格提供计算分辨率，以便在整个域内保持近似恒定的局部误差水平。这种大域/局部网格细化策略带来了一定的计算困难。首先，从深海到浅水近岸和内陆地区（包括入口、河流和周围有堤防系统的洪泛平原），水流状态的范围变化很大。不仅深度大不相同，而且描述方程中的力平衡也大不相同。在这些广泛不同的流动状态下，各种算法在稳定性、准确性和局部质量守恒特性方面表现得非常不同。其次，在局部高流量梯度和/或非常浅的水深区域提供的高水平网格分辨率实际上降低了许多算法的稳定性，这些算法在较粗的离散化中工作得很好，在具有较光滑解决方案的区域工作得很好。该项目的主要重点是通过使用适当耦合的有限元hp自适应算法来克服这些困难，这些算法基于数学上合理的误差估计。研究人员在开发用于浅水问题的连续Galerkin有限元方法方面有着广泛的历史，并且最近研究了这些问题的不连续Galerkin方法的使用。通过利用这两种方法的优势，他们计划开发用于解决浅水问题的模拟工具，这些工具可以用局部精细的非结构化网格模拟大域，可以准确地解决高梯度流动区域，可以局部适应流动特性的变化，并且在必要时遵守局部质量守恒原则。具体而言，在本项目下，研究者将：(1)进一步发展和分析二维和三维浅水流动的不连续Galerkin方法；(2)对一些模型问题的连续和不连续Galerkin方法进行全面比较；(3)研究基于两种方法耦合的浅水方程和相关数学模型的新型多算法方法。沿海海洋环流和浅水区化学物质输送的精确数学和计算机模型在经济、环境和公共卫生方面具有重要意义。主要的相互关联的问题包括沿海淹没、航行、沉积物运动、污染物运输和渔业。对飓风风暴潮的准确预测可以帮助拯救美国和全世界许多低洼地区的生命和财产。沿海潮流和水位的预测在商业和军事航行中也具有重要意义，例如在港口和航道的设计中。目前的计算机模拟工具缺乏可靠和有效地模拟这些复杂流动状态的能力。该项目的研究人员将通过使用先进的数学建模、数值算法和分布式计算技术，为这些应用开发最先进的仿真工具。

Recent progress in coastal ocean modeling has emphasized two main themes: 1) the use of relatively large computational domains which encompass much larger areas than the region of specific interest, the main concept being to place the open ocean boundaries far away in deep water non-resonant ocean basins, and 2) strategically providing computational resolution using unstructured grids in order to maintain an approximately constant level of localized error throughout the domain. This large domain/local grid refinement strategy has led to certain computational difficulties. First, the range of flow regimes varies dramatically from the deep ocean to the shallow near shore and inland regions which include inlets, rivers, and flood plains with surrounding levee systems. Not only are the depths dramatically different, but the force balances in the descriptive equations vary dramatically as well. Various algorithms perform very differently within these widely disparate flow regimes in terms of stability, accuracy and localized mass conservation properties. Second, the high level of grid resolution provided in localized high flow gradient and/or very shallow water depth regions actually degrades the stability properties of many algorithms that worked quite well with coarser discretizations, and work very well in regions with smoother solutions. The main focus of this project is to overcome these difficulties through the use of suitably coupled, finite element hp-adaptive algorithms, which are based on mathematically sound error estimates. The investigators have an extensive history in developing continuous Galerkin finite element methods for shallow water problems, and have recently investigated the use of discontinuous Galerkin methods for these problems. By exploiting the strengths of these two approaches, they plan to develop simulation tools for solving shallow water problems which can model large domains with locally refined, unstructured grids, can accurately resolve high gradient flow regions, can locally adapt to changes in flow characteristics, and which honor local mass conservation principles where necessary. Specifically, under this project, the investigators will (1) further develop and analyze discontinuous Galerkin methods for shallow water flows in two and three dimensions (2) thoroughly compare continuous and discontinuous Galerkin methods for some model problems, and (3) investigate novel multi-algorithmic approaches based on coupling the two methodologies for shallow water equations and related mathematical models.Accurate mathematical and computer modeling of coastal ocean circulation and transport of chemical species in shallow waters has significant implications from an economic, environmental and public health perspective. Major inter-related issues include coastal inundation, navigation, sediment movement, pollutant transport and fisheries. Accurate prediction of hurricane storm surges can help save lives and property in many low lying regions throughout the United States and the world. The prediction of coastal currents and water levels is also of major significance in commercial and military navigation, e.g. in the design of harbors and navigation channels. Current computer simulation tools are lacking in their ability to reliably and efficiently model these complex flow regimes. The investigators on this project, through the use of advanced mathematical modeling, numerical algorithms and distributed computing technology, will develop state-of-the-art simulation tools for these applications.