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

该项目专注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。在该项目下开发的新的数值工具将使科学家更好地了解关于KdV型方程解的数学性质的理论上尚未解决的问题。此外，该方案还将为各种应用中的非线性色散波传播的模拟提供准确和高效的数值算法。这些拟议的研究主题将对整个数学科学产生积极影响，并在许多依赖于研究非线性现象的科学领域具有重要应用。这个项目将涉及本科生和研究生，重点是让传统上在科学界代表性较低的群体的学生参与进来。通过这个项目，学生们将受益于新的算法设计的新想法，严格的数学分析方法，以及先进的实现技能。项目的目标是设计和分析第一个超收敛的可杂交不连续Galerkin(HDG)方法和用于求解KdV方程的混合混合方法及其多维推广。拟议的项目包括对新算法设计的全面覆盖，该设计得到可靠的分析支持，并通过有效的实施使其切实可行。P.I.计划对KDV型问题的超收敛HDG方法和混合混合方法进行详细的研究，主要包括以下几个步骤：第一，发展求解定常三阶线性方程的新型HDG方法和混合混合方法，重点是三阶微分算子的离散化。我们将用计算和分析的方法研究这些近似的超收敛性质。其次，P.I.希望通过使用隐式时间离散格式来求解三阶KdV方程，以避免极小的时间步长，并开发新的HDG方法和用于空间离散的混合混合方法。将进行误差分析，并研究超收敛和守恒性。第三，P.I.计划将这些超收敛方法推广到多维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.