CAREER: Development and Applications of Discontinuous Galerkin Methods

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。守恒基本原理的运用，导致了物理世界精确可靠的数学模型。这样的模型，被称为守恒定律，与相关的模型，如汉密尔顿-雅可比方程，在解中显示出丰富的特征，因此继续挑战计算科学家。本CAREER提案的主要目标是为解决这些非线性方程的高阶精确方法的发展和应用迈出几个重要的步骤。不连续伽辽金方法将构成这项工作的核心方法论，这是由于它们在准确可靠地模拟复杂问题方面具有很大的灵活性和能力。项目将全面涵盖算法设计、分析、实现和应用。该研究的成功将直接影响最优控制、图像处理、计算机视觉、天体物理学、空间物理学和能源物理学等领域问题的高效和鲁棒建模。由此产生的思想和方法将带来新的组件，以可靠地模拟科学，物理和工程中的非线性问题和复杂系统。除了训练研究生和本科生进行计算科学研究，指导女学生学习数学外，拟议的教育和学术活动还将加强区域研究小组之间的互动和合作。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The use of the basic principle of conservation has led to accurate and reliable mathematical models of the physical world. Such models, known as conservation laws, together with the related models such as Hamilton-Jacobi equations, display rich features in solutions and therefore continue to challenge the computational scientists. The primary objective of this CAREER proposal is to make several important steps towards the development and applications of high order accurate methods for solving these nonlinear equations. Discontinuous Galerkin methods will constitute the core methodology of this effort, this is due to their great flexibilities and capabilities in accurately and reliably simulating complicated problems. The project will comprehensively cover the algorithm design, analysis, implementation and applications.The success of the proposed research will have direct impact on the efficient and robust modeling of problems in areas as diverse as optimal control, image processing, computer vision, astrophysics, space physics and energy physics. The resulting ideas and methodologies will bring new components to reliably simulate nonlinear problems and complex systems arising in science, physics and engineering. Besides training graduate and undergraduate students in conducting research in computational science, mentoring women students in mathematics, the proposed educational and academic activities will also enhance interaction and collaborations among regional research groups.