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

该项目侧重于开发新的数值方法来模拟Korteweg-de Vries （KdV）型方程，该方程模拟流体力学、非线性光学、声学和等离子体物理等领域的现象。例如，KdV方程已被用于浅水波的模拟和海啸波的研究。在这个项目下开发的新的数值工具将使科学家更好地理解关于KdV型方程解的数学性质的理论上未解决的问题。此外，该项目将为各种应用中的非线性色散波传播模拟提供准确有效的数值算法。这些提出的研究课题将对整个数学科学产生积极的影响，并在许多依赖于非线性现象研究的科学领域具有重要的应用。该项目将涉及本科生和研究生，并重点关注来自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思想，严格的数学分析方法，以及先进的实现技能。本课题的目的是设计和分析求解KdV方程的第一个超收敛杂交不连续伽辽金（HDG）方法和杂交混合方法及其多维推广。提出的项目包括新算法设计的全面覆盖，该设计由可靠的分析支持，并通过有效的实现实现。本课题拟对求解KdV型问题的超收敛HDG方法和杂交混合方法进行详细研究，具体步骤如下：首先，本课题将开发求解平稳三阶线性方程的新型HDG方法和杂交混合方法，重点研究三阶微分算子的离散化问题。本文将对逼近的超收敛性质进行计算和分析研究。其次，P.I.希望通过使用隐式时间离散格式来解决三阶KdV方程，以避免极小的时间步长，并开发新的HDG方法和杂交混合方法进行空间离散。进行误差分析，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，如Kadomtsev-Petviashvili方程，而杂交技术将使这些方法在多维上有效地实现。

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.