Numerical Solution of Partial Differential Equations and Applications

许多复杂的物理现象可以用偏微分方程组(PDE)进行数学建模。例如，固体材料对施加在其上的力的响应可以用弹性方程来模拟。这些方程的解可以用于许多目的，例如预测固体结构的行为或改进其设计。弹性力学方程太复杂了，在现实情况下很难精确求解。然而，采用高级算法的高性能计算机可能会获得准确的解。这项工作的一个主要目标是开发有效的和可证明可靠的弹性解算法。在最近的突破使这种方法得以开发和验证涉及二维变形的问题之后，研究人员和他的合作者将为更困难的全三维变形情况开发类似的方法。第二个研究领域涉及求解爱因斯坦广义相对论场方程的计算机算法的发展，这是现代理解引力的基础。最近，计算作为对引力的第三种探索模式加入了理论和实验的行列，为理解方面的重大新进展奠定了基础，但也带来了巨大的挑战。广义相对论的几何和物理内容可以用偏微分方程组的语言来表示，因此，原则上可以进行数值模拟。但这只是以牺牲极其复杂的偏微分方程组为代价的，事实证明，这些系统很难在数字上求解。这里的重点将是理解与爱因斯坦方程的数值解相关的基本问题。这项工作具有更广泛的影响。在许多具有挑战性的工业和工程应用中，例如飞机、高级建筑和桥梁以及海上石油平台，都需要稳健可靠的方法来求解弹性方程。最近的设计失败，其中一些是灾难性的，被追溯到弹性的数值算法不足。因此，该项目有可能为公共安全和繁荣做出贡献。广义相对论的计算算法被认为是引力天文学这一新兴基础科学成功的关键，这是该项目的另一个重大潜在影响。这项工作中开发的技术也有望应用于其他重要的PDE系统。此外，该项目将为科学的基础设施做出贡献。它将通过合作和研究人员将在项目期间组织的讲习班，推动建立一个有引力天体物理学数学家参与的跨学科社区。调查员将通过出版物、会议和研讨会报告以及万维网广泛传播研究成果。最后，该项目将通过让学生和博士后参与重要的、前沿的、跨学科的研究，直接支持年轻科学家的培训和科学广度。

Many complex physical phenomena can be modeled mathematically by systems of partial differential equations (PDEs). For example, the response of a solid material to forces imposed on it can be modeled by the equations of elasticity. Solutions of these equations can then be used for many purposes, such as to predict the behavior of a solid structure or improve its design. The equations of elasticity are far too complex to solve exactly in realistic situations. However, high performance computers with advanced algorithms may obtain accurate solutions. A major goal of this work is to develop efficient and certifiably reliable solution algorithms for elasticity. Following on recent breakthroughs which enabled the development and certification of such methods for problems involving two-dimensional deformations, the investigator and his collaborators will develop analogous methods for the much more difficult situation of full three-dimensional deformations.The second area of investigation concerns the development of computer algorithms for solving Einstein's field equations of general relativity, which are the basis for the modern understanding of gravity. Computation has recently joined theory and experiment as a third mode of inquiry into gravity, setting the stage for major new advances in understanding, but also bringing tremendous challenges. The geometric and physical content of general relativity can be expressed in the language of PDEs, and thus, in principle, made amenable to numerical simulation. But this is achieved only at the expense of extremely complex systems of PDEs, which have proven very difficult to solve numerically. The emphasis here will be on understanding the fundamental issues relevant to the numerical solution of the Einstein equations in cases of physical interest.This work has many broader impacts. Robust and reliable methods for solving the equations of elasticity are needed in many challenging industrial and engineering applications, for example for aircraft, advanced buildings and bridges, and offshore oil platforms. Recent design failures, some of them catastrophic, have been traced to inadequate numerical algorithms for elasticity. Thus this project has the potential to contribute to public safety and prosperity. Computational algorithms for general relativity are recognized as crucial to the success of the emerging fundamental science of gravitational astronomy, another large potential impact of this project. The techniques developed in this work are expected to apply to other important systems of PDEs as well. Moreover, the project will contribute to the infrastructure of science. It will advance the development of an interdisciplinary community involving mathematicians in gravitational astrophysics, both through collaborations and through a workshop that the investigator will organize during the project period. The investigator will disseminate the results of the research broadly through publication, conference and seminar presentations, and the world wide web. Finally, the project will directly support the training and scientific breadth of young scientists by involving students and postdocs in important, cutting-edge, interdisciplinary research.