Mathematical Sciences: Numerical Methods & Conservation Laws

研究人员开发了多维高分辨率有限体积方法来求解非线性双曲型守恒律方程组以及在各种应用中出现的相关问题。他开发的公共领域软件包CLAWPACK(守恒律程序包)被扩展到处理一维、二维和三维空间中笛卡尔网格和曲线网格上的更广泛的问题。进一步发展了镶嵌复合网格，以允许不规则边界附近的贴体网格与远离边界的笛卡尔网格相耦合。浸没界面方法用于处理溶液或其导数中的不连续性，并与多维守恒律方法相结合来求解具有材料界面的流体动力学和波传播问题。这些技术在被不规则界面切割的均匀笛卡尔网格上实现了二阶精度。类似的技术也适用于化学反应或燃烧引起的源项较硬的问题，导致薄的反应区在宏观上表现为界面。研究了一些领域的具体应用，包括地下水流动、大气流动、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，包括大量的文件和应用实例。与之配套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，用于解决几乎所有科学和工程领域中出现的一类数学问题。所考虑的偏微分方程组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学流动、地下水或地球表面下的石油)，或流体或空气中的波(例如空气或海洋中的声波或物体的超声波勘探中的声波、地球上地震产生的地震波或为石油勘探而人工产生的地震波、雷达波)。甚至生态模型中的有机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年来的广泛研究，主要是在空气动力学和武器开发界。这项技术正在慢慢地转移到其他领域，但大多数算法的复杂性阻碍了这项技术的发展。这个项目的软件应该有助于加快这一进程。它的设计目的是作为教学和研究工具的普遍使用，在许多应用领域包括大量的例子。还开发了新的方法来处理在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耦合)，以及在几何复杂的空间区域中的问题，这些区域由不规则边界或包含材料特性变化的界面(例如，地下水流动和地震中的不同类型的岩石之间，或超声成像中的骨骼和组织之间)所包围。目前正在与许多领域的研究人员密切合作(特别是地下水流动和大气模拟)，以改进和推广该软件，并将其用于解决具体问题。地下水中污染物的迁移是一个非常有意义的应用，在具有不连续渗透率和不规则几何形状的多孔介质中，线性或非线性平流经常必须与用于吸附和反应的刚性源项相耦合。准确的模型既有助于修复受污染的场地，也有助于研究拟议的核废料地下储存点。大气模型在短期天气预报和气候、臭氧消耗等长期全球模型中都是至关重要的。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。

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.