Adaptive multinumeric finite element methods for shallow water flow

沿海海洋模拟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，其覆盖的区域比特定感兴趣的区域大得多，主要概念是将开放的海洋边界放置在远离深水非共振洋盆的位置，以及2)战略性地使用非结构网格提供计算分辨率，以便在整个区域保持大约恒定的局部误差水平。这种大域/局部网格细化策略导致了一定的计算困难。首先，水流流态的范围从深海到浅海近岸和内陆地区都有很大的不同，包括进水口、河流和带有周围堤防系统的泛滥平原。不仅深度有很大的不同，描述方程中的力平衡也有很大的不同。在稳定性、准确性和局部质量守恒性质方面，不同的算法在这些完全不同的流型中执行非常不同的操作。其次，在局部高流动梯度和/或极浅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下工作得很好，在解较光滑的区域工作得很好。这个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。研究人员在发展浅水问题的连续Galerkin有限元方法方面有很长的历史，最近又研究了使用间断Galerkin方法来解决这些问题。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区域，可以准确地解析高梯度流动区域，可以局部适应流动特征的变化，并在必要时遵守局部质量守恒原则。具体地说，在该项目下，研究人员将(1)进一步发展和分析二维和三维浅水流动的间断Galerkin方法(2)深入比较一些模型问题的连续Galerkin方法和间断Galerkin方法，(3)研究基于这两种方法耦合的浅水方程和相关数学模型的新的多算法方法。从经济、环境和公共卫生的角度对沿海海洋循环和浅水化学物质输运的精确数学和计算机模拟具有重要意义。主要相互关联的问题包括沿海淹没、航行、泥沙运动、污染物运输和渔业。准确预测飓风风暴潮可以帮助拯救美国和世界许多低洼地区的生命和财产。沿岸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型的能力。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模拟工具。

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.