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.

该项目集中于开发新型计算方法，以有效地解决具有保守物理特性或高度振荡波解决方案的问题。新的保守方法可以保留对身体感兴趣的数量，并在长期内允许准确稳定的模拟。它们将用于各种领域的应用，例如流体动力学，非线性光学元件，等离子体物理学和Bose-Einstein冷凝物。新的多尺度方法可以准确有效地捕获高度振荡的波解决方案。他们将对量子力学的研究产生积极的影响，并在应用超快和低消耗纳米级电子设备的设计方面具有巨大的潜力。项目中开发的方法将帮助人们理解理论上未解决的问题，并为设计竞争性数值算法提供新的框架，以解决其他复杂问题。该项目还将涉及对本科生和研究生的指导和培训，包括传统代表性不足的群体。它将为学生提供充分的机会将研究整合到他们的教育经验中。该项目包括以下主题：（1）对新颖的保守性不连续的Galerkin（DG）方法的深入调查以及对Korteweg-de Vries（KDV）方程（KDV）方程（KDV）方程的隐性惩罚参数，（2）通过对保守性DG方法进行保守的dg损失模型的HER HER，包括HEH HER，包括Korteweg-de Vries（KDV）方程（KDV）方程（KDV）方程（KDV）方程（2） 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捕获粗网格上Schrodinger方程的高度振荡溶液。前两个主题中的新颖思想是通过隐式惩罚来执行保护特性，这可以推广到具有物理量保存的其他类型的问题。 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.