Development of superconvergent hybridizable discontinuous Galerkin methods and mixed methods for Korteweg-de Vries type equations

超收敛杂化间断伽辽金方法和 Korteweg-de Vries 型方程混合方法的发展

• 批准号: 1419029
• 负责人: Bo Dong
• 金额: $ 12.99万
• 依托单位: University of Massachusetts, Dartmouth
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-08-15 至 2018-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1419029&HistoricalAwards=false
• 关键词: Development; superconvergent; hybridizable; discontinuous; Galerkin

The project focuses on developing novel numerical methods for simulating the Korteweg-de Vries (KdV) type equations, that model phenomena in areas such as fluid mechanics, nonlinear optics, acoustics, and plasma physics. For example, the KdV equation has been used in the modeling of shallow water waves and the study of Tsunami waves. The new numerical tools developed under this project will provide scientists with a better understanding of theoretically unresolved issues on the mathematical properties of solutions to KdV type equations. Furthermore, the proposed project will provide accurate and efficient numerical algorithms for the simulation of nonlinear dispersive wave propagation in various applications. These proposed research topics will have a positive impact across the mathematical sciences and have significant applications in many scientific areas that rely on the study of non-linear phenomena. This project will involve undergraduate and graduate students and focus on involving student from groups traditionally underrepresented in the sciences. By working on the project, the students will benefit from novel ideas for new algorithm design, approaches for rigorous mathematical analysis, and advanced skills in implementation.The objective of the project is to devise and analyze the first superconvergent hybridizable discontinuous Galerkin (HDG) methods and hybridized mixed methods for solving the KdV equations and their multidimensional generalizations. The proposed project includes a comprehensive coverage of new algorithm design that is backed up by solid analysis and made practical by efficient implementation. The P.I. proposes to carry out a detailed study of superconvergent HDG methods and hybridized mixed methods for KdV type problems in the following steps: First, the P.I. will develop novel HDG methods and hybridized mixed methods for stationary third-order linear equations, focusing on the discretization of the third-order differential operator. Superconvergence properties of the approximations will be computationally and analytically investigated. Second, the P.I. would like to solve the third-order KdV equations by using implicit schemes for time discretization to avoid extremely small time steps and developing new HDG methods and hybridized mixed methods for spatial discretization. Error analysis will be carried out, and superconvergence and conservativity properties will be studied. Third, the P.I. plans to extend these superconvergent methods to multidimensional KdV type equations such as the Kadomtsev-Petviashvili equation, and the hybridization technique will make the methods efficiently implementable in multiple dimensions.

该项目的重点是开发用于模拟 Korteweg-de Vries (KdV) 类型方程的新颖数值方法，该方程对流体力学、非线性光学、声学和等离子体物理等领域的现象进行建模。例如，KdV方程已用于浅水波的建模和海啸波的研究。该项目开发的新数值工具将使科学家更好地理解有关 KdV 型方程解的数学性质的理论上尚未解决的问题。此外，该项目将为各种应用中非线性色散波传播的模拟提供准确有效的数值算法。这些提出的研究主题将对整个数学科学产生积极影响，并在许多依赖非线性现象研究的科学领域具有重要应用。该项目将涉及本科生和研究生，并重点关注传统上在科学领域代表性不足的群体的学生。通过参与该项目，学生将受益于新算法设计的新颖想法、严格的数学分析方法以及高级实施技能。该项目的目标是设计和分析第一个超收敛混合不连续伽辽金（HDG）方法以及用于求解 KdV 方程及其多维推广的混合混合方法。 拟议的项目全面涵盖了新算法设计，该设计有可靠的分析支持，并通过有效的实施而变得实用。 P.I.提出对KdV类问题的超收敛HDG方法和杂化混合方法进行详细研究，步骤如下：首先，P.I.将针对平稳三阶线性方程开发新颖的 HDG 方法和混合混合方法，重点关注三阶微分算子的离散化。近似值的超收敛性质将通过计算和分析进行研究。其次，P.I.希望通过使用隐式时间离散方案来求解三阶 KdV 方程以避免极小的时间步长，并开发新的 HDG 方法和混合混合方法进行空间离散。将进行误差分析，并研究超收敛性和保守性。第三，P.I.计划将这些超收敛方法扩展到多维 KdV 型方程，例如 Kadomtsev-Petviashvili 方程，而混合技术将使这些方法在多个维度上有效实现。