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

具有隐式惩罚参数的保守间断伽辽金方法和偏微分方程的多尺度可杂交间断伽辽金方法

• 批准号: 2309670
• 负责人: Bo Dong
• 金额: $ 36.54万
• 依托单位: University of Massachusetts, Dartmouth
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-06-15 至 2026-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2309670&HistoricalAwards=false
• 关键词: Conservative; discontinuous; Galerkin; methods; implicit

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.

本项目致力于开发新的计算方法，以有效地解决具有保守物理性质或高度振荡波解的问题。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准确和稳定的模拟。它们在流体力学、非线性光学、等离子体物理和玻色-爱因斯坦凝聚等领域有着广泛的应用。新的多尺度方法可以准确和有效地捕捉高振荡波的解决方案。它们将对量子力学的研究产生积极的影响，并在超快和低功耗纳米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。该项目还将涉及对本科生和研究生的辅导和培训，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体验的绝佳机会。该项目包括以下主题：（1）深入研究了Korteweg-de弗里斯（KdV）方程的新型隐式罚参数守恒间断Galerkin（DG）方法，（2）发展了适用于具有守恒性质的更复杂波动模型的隐式罚参数守恒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、可杂交不连续Galerkin（HDG）方法与多尺度的基础上有效地捕捉高振荡的解决方案薛定谔方程的粗网格。前两个主题中的新颖思想是通过隐式惩罚来加强守恒性质，这可以推广到其他类型的物理量守恒问题。第三个主题中的方法集成了高效的HDG框架和多尺度非多项式基函数，这使得它们在粗网格和细网格上都比传统的薛定谔方程有限元方法表现得更好。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。