Conservative discontinuous Galerkin methods with implicit penalty parameters and multiscale hybridizable discontinuous Galerkin methods for PDEs

该项目专注于开发新的计算方法，以有效地解决具有守恒物理性质或高振荡波解的问题。新的保守方法可以保留物理上感兴趣的量，并允许在很长一段时间内精确和稳定的模拟。它们将在流体动力学、非线性光学、等离子体物理和玻色-爱因斯坦凝聚等各个领域得到应用。新的多尺度方法可以准确有效地捕获高振荡波解。它们将对量子力学的研究产生积极的影响，并在超快和低消耗的纳米级电子器件的设计中具有巨大的应用潜力。该项目开发的方法将帮助人们理解理论上尚未解决的问题，并为设计解决其他复杂问题的竞争性数值算法提供新的框架。该项目还将涉及指导和培训本科生和研究生，包括传统上代表性不足的群体。它将为学生提供将研究融入他们的教育经历的绝佳机会。该项目包括以下主题：(1)深入研究了具有隐式惩罚参数的Korteweg-de Vries （KdV）方程的新型保守不连续Galerkin （DG）方法；(2)通过隐式惩罚发展了具有守恒性质的更复杂波动模型的保守DG方法，包括Hirota-Satsuma耦合KdV系统、Schrodinger-KdV系统、abcd-Boussinesq系统以及二维Zakharov-Kuznetsov （ZK）方程和Kadomtsev-Petviashvili （KP）方程；(3)设计、分析和实现了基于多尺度基的可杂交不连续伽辽金（HDG）方法，用于在粗网格上有效捕获薛定谔方程的高振荡解。前两个主题中的新思想是通过隐式惩罚来强制执行守恒性质，这可以推广到其他类型的以物理量守恒为特征的问题。第三主题的方法将高效HDG框架和多尺度非多项式基函数相结合，使其在粗网格和细网格上都优于传统的薛定谔方程有限元方法。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project concentrates on the development of novel computational methods for efficiently solving problems that have conserved physical properties or highly oscillatory wave solutions. The new conservative methods can preserve physically interested quantities and allow accurate and stable simulations over a long time period. They will be useful for applications in various fields, such as fluid dynamics, nonlinear optics, plasma physics, and Bose-Einstein condensates. The new multiscale methods can accurately and efficiently capture highly oscillatory wave solutions. They will have a positive impact in the study of quantum mechanics and great potential in application to the design of ultrafast and low consumption nanoscale electronic devices. The methods developed in the project will help people understand theoretically unresolved issues and provide new frameworks for devising competitive numerical algorithms for solving other complex problems. The project will also involve mentoring and training of undergraduate and graduate students, including the traditionally underrepresented groups. It will provide students great opportunities to integrate research into their educational experience.The project includes the following topics: (1) in-depth investigation of the novel conservative discontinuous Galerkin (DG) method with implicit penalty parameters for the Korteweg-de Vries (KdV) equation, (2) development of conservative DG methods via implicit penalization for more complicated wave models with conservation properties, including the Hirota-Satsuma coupled KdV system, the Schrodinger-KdV system, the abcd-Boussinesq system, and the two-dimensional Zakharov-Kuznetsov (ZK) equation and Kadomtsev-Petviashvili (KP) equation, (3) design, analysis, and implementation of hybridizable discontinuous Galerkin (HDG) methods with multiscale basis for efficiently capturing highly oscillatory solutions of Schrodinger equations on coarse meshes. The novel idea in the first two topics is to enforce conservation properties via implicit penalization, and this can be generalized to other types of problems that feature conservation of physical quantities. The methods in the third topic integrate the efficient HDG framework and the multiscale non-polynomial basis functions, which makes them perform better than traditional finite element methods for Schrodinger equations on both coarse meshes and fine meshes.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.