Parallel, Adaptive Cartesian Grid Algorithms for Natural Hazards Modeling

洪水、海啸、泥石流、山体滑坡、风暴潮和潜在的大坝溃坝等地球物理灾害威胁着美国乃至全球的社区。该项目开发了能够有效模拟这些灾害的计算工具，使不同的研究人员和应急规划人员能够为最有可能受到这些灾害影响的地区绘制灾害地图。为了提高效率，计算框架使用自适应的“深度平均”数学模型，只需要二维平面网格，而不是完全的三维网格。该项目的一个主要目标是校正我们的深度平均模型，以捕获可能溢出防洪屏障或海港防波堤的局部波浪。这些校正项将为计算框架的当前用户提供模拟浅层地球物理灾害的关键附加功能，并允许他们创建更可靠的危险地图。计算工具还可以充分利用桌面工作站、中等规模的计算集群以及NSF资助的超级计算站点上可用的大规模并行计算设施上出现的硬件趋势。该项目还为用户提供了使用开源软件（如谷歌Earth浏览器）可视化结果的工具。最终，计算建模可以帮助响应者预测在发生不可避免的危险时如何分配应急资源，并向开发商、立法代表和公民通报其社区中的潜在风险。本文将重点研究在自适应精四叉树网格上实现变系数椭圆问题的直接求解器。目标求解器是由A. Gillman和P. Martinsson开发的分层庞加莱Steklov （HPS）求解器。该求解器满足四个关键特性，(1)具有无矩阵方法的易用性，(2)可以快速求解附近系统，(3)具有最优的O(N)效率，(4)提供可调的参数，以按精度要求比例降低计算成本。此外，该方法利用低秩近似压缩密集矩阵，加快矩阵计算速度。在提出的工作中，PI将修改原始的HPS求解器，以使用二阶有限体积方案，并在ForestClaw中实现求解器，这是一种并行的、基于补丁的笛卡尔自适应四叉树代码。PI还将报告HPS方法实现的可扩展性和并行效率。将出现的两个技术挑战是开发跨处理器边界合并Dirichlet-to-Neumann映射的有效程序，以及增量更新动态演化网格的求解器分解。目标应用是求解Serre-Green Naghdi方程，用于模拟浅水波动方程的色散修正。这些更正将包括在ForestClaw的GeoClaw扩展。GeoClaw （D. George， R. J. LeVeque， M. J. Berger）是一个广泛使用的求解深度平均流动方程的软件包。在GeoClaw扩展中添加这些校正术语将为GeoClaw用户提供海啸、洪水、泥石流、风暴潮和其他浅层地球物理流动建模的关键功能。最终，所提出的求解器可以在ForestClaw框架内作为各种物理流动现象的通用椭圆求解器使用。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Geophysical hazards such as flooding, tsunamis, debris flows, landslides, storm surges and potential dam failures threaten communities across the United States and globally. This project develops computational tools that can efficiently simulate these hazards, enabling a diverse group of researchers and emergency planners to develop hazard maps of areas most likely to be impacted by these disasters. For efficiency, the computational framework uses adaptive, "depth-averaged" mathematical models that only require two dimensional planar grids, rather than fully three dimensional meshes. A primary goal of the project is to correct our depth-averaged model to capture localized waves that may spill over flood barriers or overtop harbor breakwaters. These correction terms will give current users of the computational framework critical additional capabilities for modeling shallow geophysical hazards and allow them to create more robust hazard maps. The computational tools can also take full advantage of emerging hardware trends available on desktop workstations, moderate sized compute clusters, as well as massively parallel computing facilities available at NSF funded supercomputing sites. The project also provides users with tools for visualizing results using open source software such as the Google Earth browser. Ultimately, computational modeling can aid responders in predicting how to distribute emergency resources in the event of unavoidable hazards and serve to inform developers, legislative representatives, and citizenry of potential risks in their communities.The research will focus on the implementation of a direct solver for variable coefficient elliptic problems on adaptively refined quad-tree meshes. The targeted solver is the Hierarchical Poincare Steklov (HPS) solver, developed by A. Gillman and P. Martinsson. Satisfying four crucial properties, this solver (1) has the ease of use of matrix-free methods, (2) can solve nearby systems quickly, (3) has optimal O(N) efficiency, and (4) provides parameters that can be tuned to reduce computational cost in proportion to accuracy requirements. Furthermore, the method uses low rank approximations to compress dense matrices and accelerate matrix computations. In the proposed work, the PI will modify the original HPS solver for use with second order, finite volume schemes and implement the solver in ForestClaw, the parallel, patch-based Cartesian adaptive quad-tree code. The PI will also report on the scalability and parallel efficiency of the implementation of the HPS method. Two technical challenges that will arise are to develop effective procedures for merging Dirichlet-to-Neumann maps across processor boundaries and incrementally updating the solver factorization for dynamically evolving meshes. The targeted application is the solution to the Serre-Green Naghdi equations for modeling dispersive corrections to the shallow water wave equations. These corrections will be included in the GeoClaw extension of ForestClaw. GeoClaw (D. George, R. J. LeVeque, M. J. Berger) is a widely used software package for solving depth-averaged flow equations. The addition of these correction terms to the GeoClaw extension will provide GeoClaw users with critical capabilities for modeling tsunamis, flooding, debris flows, storm surges and other shallow geophysical flows. Ultimately, the proposed solver can be used within the ForestClaw framework as a general purpose elliptic solver for a variety of physical flow phenomena.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.