Mathematical Sciences: Numerical Methods & Conservation Laws

数学科学：数值方法

• 批准号: 9505021
• 负责人: Randall LeVeque
• 金额: $ 19.39万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-08-01 至 1999-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9505021&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Numerical; Methods; &amp

The investigator develops multi-dimensional high resolution finite volume methods for solving nonlinear hyperbolic systems of conservation laws and related problems arising in a variety of applications. The public domain software package CLAWPACK (Conservation LAWs PACKage) he has developed is extended to handle a wider variety of problems on both Cartesian and curvilinear grids in 1, 2, and 3 space dimensions. Mosaic composite grids are further developed to allow body-fitted grids near an irregular boundary to be coupled with Cartesian grids away from the boundary. Immersed interface methods for handling discontinuities in the solution or its derivatives are used in conjunction with multi-dimensional conservation law methods to solve fluid dynamics and wave propagation problems with material interfaces. These techniques achieve second order accuracy on uniform Cartesian grids cut by irregular interfaces. Similar techniques are applied to problems with stiff source terms arising from chemical reactions or combustion, giving rise to thin reaction zones that behave macroscopically as interfaces. Specific applications in a number of areas are studied, including groundwater flow, atmospheric flow, chemotaxis, and astrophysics. The software being developed is intended for teaching as well as research purposes, and includes extensive documentation and applied examples. An accompanying textbook is being written. The investigator develops computational methods and public domain software for the solution of a class of mathematical problems that arises in virtually every field of science and engineering. The partial differential equations considered can, in various forms, model the motion of liquids or gas (e.g., air in the atmosphere, water in the ocean, aerodynamic flow around aircraft or through turbines, groundwater or oil beneath the earth's surface), or the motion of waves in fluid or air (e.g., acoustic waves in the air or ocean or in ultrasonic explor ation of the body, seismic waves in the earth originating from earthquakes or artificially generated for oil exploration, radar waves). Even the motion of organisms in ecological modeling or cells in developmental biology follows similar laws. The methods are based on extensive research over the past 20 years, primarily in the aerodynamics and weapons development communities. This technology is slowly being transferred to other areas, but is hindered by the complexity of most of the algorithms. The software of this project should help speed this process. It is designed for general use as both a teaching and research tool, with extensive examples included in many applications areas. Novel methods are also developed to deal with phenomena occurring on different time scales (e.g., fast chemical reactions coupled with slow groundwater or atmospheric flow) and for problems in geometrically complicated regions of space bounded by irregular boundaries or containing interfaces where material properties change (e.g., between different types of rock in groundwater flow and seismology, or between bone and tissue in ultrasound imaging). Close collaboration is underway with researchers in many areas (particularly groundwater flow and atmospheric modeling), both to improve and generalize the software and to use it in the solution of specific problems. An application of great interest is contaminant transfer in groundwater flow, where linear or nonlinear advection in a porous medium with discontinuous permeabilities and irregular geometries must often be coupled with stiff source terms for adsorption and reactions. Accurate models are needed both as an aid to remediation of polluted sites and to the study of proposed underground storage sites for nuclear waste. Atmospheric modeling is crucial both in short-term weather prediction and in long-range global modeling of climate, ozone depletion, etc. The investigator works with researchers in these areas to incorporate this software into standard models as well as to develop new methods where needed.

研究者开发了多维高分辨率有限体积方法，用于解决在各种应用中产生的非线性双曲线系统和相关问题。 他开发的公共域软件包ClawPack（保护法律软件包）已扩展，以在1、2和3空间维度的笛卡尔和曲线网格上处理各种各样的问题。 进一步开发了镶嵌复合网格，以允许在不规则边界附近的车身拟合网格与远离边界的笛卡尔网格结合。 混合界面方法用于处理溶液或其衍生物中的不连续性的方法与多维保护定律方法结合使用，以解决材料接口的流体动力学和波传播问题。 这些技术在不规则界面切开的均匀笛卡尔网格上达到了二阶精度。 类似的技术应用于化学反应或燃烧引起的僵硬术语的问题，从而引起薄反应区，这些反应区以宏观的形式作为接口。 研究了许多区域中的特定应用，包括地下水流，大气流，趋化性和天体物理学。 正在开发的软件旨在进行教学和研究目的，其中包括大量文档和应用示例。 正在写一本随附的教科书。 研究者开发了计算方法和公共领域软件，以解决几乎每个科学和工程领域中出现的一类数学问题的解决方案。 The partial differential equations considered can, in various forms, model the motion of liquids or gas (e.g., air in the atmosphere, water in the ocean, aerodynamic flow around aircraft or through turbines, groundwater or oil beneath the earth's surface), or the motion of waves in fluid or air (e.g., acoustic waves in the air or ocean or in ultrasonic explor ation of the body, seismic waves in the earth originating from earthquakes or人为地生成用于油勘探的，雷达波）。 即使是生态建模或发育生物学中细胞中生物体的运动也遵循类似的定律。 这些方法基于过去20年的广泛研究，主要是在空气动力和武器开发社区中。 这项技术正在慢慢转移到其他领域，但受到大多数算法的复杂性的阻碍。 该项目的软件应有助于加快此过程。 它旨在作为一般用作作为教学和研究工具，在许多应用领域中包含了广泛的示例。 还开发了新的方法来处理在不同的时间尺度上发生的现象（例如，快速化学反应与缓慢的地下水或大气流相结合），以及在几何复杂的空间区域中的问题，包括不规则边界的空间区域或包含材料特性的接口（例如，在地下水流动和骨之间的不同类型的岩石中，材料特性都会变化（例如，在地下水和骨之间的不同类型）中，或者是在骨之间或组织中的组织。 与许多领域的研究人员（尤其是地下水流量和大气建模）正在进行密切合作，以改善和推广软件并将其用于解决特定问题的解决方案。 极大的兴趣应用是地下水流中的污染物转移，在地下水流中，线性或非线性对流在具有不连续的渗透率和不规则几何形状的多孔培养基中，通常必须与坚硬的源术语结合以进行吸附和反应。 需要准确的模型既有帮助来修复污染地点，又需要对拟议的地下存储地点进行核废料的研究。 在短期天气预测和气候，臭氧耗竭等的远程全球建模中，大气建模至关重要。研究人员与这些领域的研究人员合作，将此软件纳入标准模型，并在需要的情况下开发新方法。