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

该项目的重点是开发新的数值方法来模拟Korteweg-de弗里斯（KdV）型方程，该模型在流体力学，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的模拟提供准确有效的数值算法。这些拟议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中具有重要的应用。该项目将涉及本科生和研究生，重点关注传统上在科学领域代表性不足的群体的学生。通过该项目的工作，学生将受益于新算法设计的新思想，严格的数学分析方法和先进的实现技能。该项目的目标是设计和分析第一个超收敛的杂交间断Galerkin（HDG）方法和杂交混合方法来求解KdV方程及其多维推广。 拟议的项目包括一个新的算法设计的全面覆盖，是由坚实的分析和有效的实施切实可行的备份。私家侦探提出了对KdV型问题的超收敛HDG方法和杂交混合方法进行详细研究的步骤：首先，P.I.将发展新的HDG方法和杂交混合方法用于定常三阶线性方程组，重点是三阶微分算子的离散化。超收敛性质的近似将计算和分析研究。第二，PI。希望通过使用隐式格式进行时间离散以避免极小的时间步长，并发展新的HDG方法和杂交混合方法进行空间离散来求解三阶KdV方程。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，如Kadomtsev-Petviashvili方程，杂交技术将使这些方法在多维中有效地实现。

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.