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.

研究人员开发了多维高分辨率有限体积方法，用于求解守恒定律的非线性双曲系统以及各种应用中出现的相关问题。 他开发的公共领域软件包 CLAWPACK（守恒定律包）经过扩展可以处理 1、2 和 3 空间维度的笛卡尔和曲线网格上的更广泛的问题。 马赛克复合网格进一步发展，允许靠近不规则边界的贴体网格与远离边界的笛卡尔网格耦合。 用于处理解或其导数中的不连续性的浸入界面方法与多维守恒定律方法结合使用，以解决材料界面的流体动力学和波传播问题。 这些技术在由不规则界面切割的均匀笛卡尔网格上实现了二阶精度。 类似的技术适用于化学反应或燃烧引起的刚性源项问题，从而产生宏观上表现为界面的薄反应区。 研究了许多领域的具体应用，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用于教学和研究目的，并包括大量文档和应用示例。 随附的教科书正在编写中。 研究人员开发计算方法和公共领域软件来解决几乎在科学和工程的每个领域中出现的一类数学问题。 所考虑的偏微分方程可以以各种形式模拟液体或气体的运动（例如，大气中的空气、海洋中的水、飞机周围或通过涡轮机的空气动力流、地球表面下的地下水或石油），或者流体或空气中波的运动（例如，空气或海洋中的声波或人体超声波探测中的声波、地球中源自地震或为石油勘探而人工产生的地震波、雷达波）。 甚至生态模型中的生物体或发育生物学中的细胞的运动也遵循类似的规律。 这些方法基于过去 20 年的广泛研究，主要是在空气动力学和武器开发领域。 这项技术正在慢慢转移到其他领域，但受到大多数算法复杂性的阻碍。 该项目的软件应该有助于加快这一过程。 它被设计为通用的教学和研究工具，在许多应用领域都包含大量示例。 还开发了新的方法来处理不同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动相结合），以及解决由不规则边界或包含材料属性变化的界面所界定的几何复杂空间区域中的问题（例如， ，地下水流和地震学中不同类型的岩石之间，或超声成像中的骨骼和组织之间）。 我们正在与许多领域（特别是地下水流和大气建模）的研究人员进行密切合作，以改进和推广该软件，并将其用于解决特定问题。 一个令人感兴趣的应用是地下水流中的污染物转移，其中具有不连续渗透率和不规则几何形状的多孔介质中的线性或非线性平流通常必须与吸附和反应的刚性源项耦合。 需要准确的模型来帮助修复污染场地和研究拟议的核废料地下储存场。 大气建模对于短期天气预报和气候、臭氧消耗等的长期全球建模都至关重要。研究人员与这些领域的研究人员合作，将该软件纳入标准模型，并在需要时开发新方法。