Multiscale and Hybridizable Discontinuous Galerkin Methods for Dispersive Equations and Systems

色散方程和系统的多尺度和可混合非连续伽辽金方法

• 批准号: 1818998
• 负责人: Bo Dong
• 金额: $ 26.92万
• 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: 2018
• 资助国家: 美国
• 起止时间: 2018-07-01 至 2022-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1818998&HistoricalAwards=false
• 关键词: Multiscale; Hybridizable; Discontinuous; Galerkin; Methods

This project concentrates on the development of novel computational methods for efficiently solving dispersive equations and systems, including the time-independent Schrodinger equations and Korteweg-de Vries (KdV) type equations in multidimensional spaces and systems. Schrodinger equations play a central role in the study of quantum mechanical systems and are widely used in the simulation of quantum transport in nanoscale semiconductor devices. The proposed multiscale method for Schrodinger equations will have a positive impact in the study of quantum mechanics and great potential in applications to ultrafast, low consumption and high functionality nanoscale semiconductor devices. KdV type equations and systems have wide applications in various fields such as fluid mechanics, nonlinear optics, acoustics, plasma physics, and Bose-Einstein condensates. The proposed method for KdV type equations and systems will help understand theoretically unresolved issues and provide accurate and efficient numerical tools for simulation of nonlinear waves in applications. The proposed research includes the following topics, (1) development and analysis of multiscale discontinuous Galerkin (DG) methods for Schrodinger equations in 1D, system, and 2D for the simulation of nanoscale semiconductor structures on coarse meshes, (2) design and error analysis of hybridizable discontinuous Galerkin (HDG) methods for solving multidimensional KdV type equations and KdV type systems, and (3) design of IMEX HDG-DG schemes for efficiently solving KdV type nonlinear equations and systems. To efficiently resolve highly oscillatory solutions of Schrodinger equations on coarse meshes, the multiscale DG methods will incorporate the oscillatory nature of the solutions and thus the multiple scales into the non-polynomial basis functions. For KdV type equations in multi-dimensions and systems, the PI will devise new HDG methods, study their convergence and conservation properties, and combine them with other DG methods in IMEX scheme to achieve high-order solutions both in time and in space and avoid overly small time-step sizes.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.

该项目致力于开发有效求解色散方程和系统的新型计算方法，包括多维空间和系统中的时间无关薛定谔方程和 Korteweg-de Vries (KdV) 型方程。 薛定谔方程在量子力学系统的研究中发挥着核心作用，并广泛应用于纳米级半导体器件中量子输运的模拟。所提出的薛定谔方程的多尺度方法将对量子力学的研究产生积极的影响，并在超快、低功耗和高功能纳米半导体器件的应用中具有巨大的潜力。 KdV型方程和系统在流体力学、非线性光学、声学、等离子体物理、玻色-爱因斯坦凝聚体等各个领域有着广泛的应用。所提出的 KdV 型方程和系统方法将有助于理解理论上未解决的问题，并为应用中的非线性波模拟提供准确有效的数值工具。 拟议的研究包括以下主题，（1）开发和分析一维、系统和二维薛定谔方程的多尺度不连续伽辽金（DG）方法，用于在粗网格上模拟纳米级半导体结构，（2）设计和误差分析用于求解多维 KdV 型方程和 KdV 型系统的可混合间断伽辽金 (HDG) 方法，以及 (3) 用于高效求解的 IMEX HDG-DG 方案的设计KdV型非线性方程和系统。为了有效地求解粗网格上薛定谔方程的高振荡解，多尺度 DG 方法将结合解的振荡性质，从而将多个尺度纳入非多项式基函数中。对于多维、多系统的KdV型方程，PI将设计新的HDG方法，研究其收敛性和守恒性质，并将其与IMEX方案中的其他DG方法相结合，以实现时间和空间上的高阶解，并避免时间步长过小。该奖项反映了 NSF 的法定使命，并且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。

项目成果

Numerical Investigations on the Resonance Errors of Multiscale Discontinuous Galerkin Methods for One-Dimensional Stationary Schrödinger Equation

一维平稳薛定谔方程多尺度间断伽辽金法共振误差的数值研究

• DOI: 10.1007/s42967-022-00248-4
• 发表时间: 2023-03
• 期刊: Communications on Applied Mathematics and Computation
• 影响因子: 1.6
• 作者: Dong, Bo;Wang, Wei
• 通讯作者: Wang, Wei

A high-order multiscale discontinuous Galerkin method for two-dimensional Schrödinger equation in quantum transport

量子输运中二维薛定谔方程的高阶多尺度间断伽辽金法

• DOI: 10.1016/j.cam.2022.114701
• 发表时间: 2023-01
• 期刊: Journal of Computational and Applied Mathematics
• 影响因子: 2.4
• 作者: Dong, Bo;Wang, Wei
• 通讯作者: Wang, Wei

High-order multiscale discontinuous Galerkin methods for the one-dimensional stationary Schrödinger equation

一维平稳薛定谔方程的高阶多尺度间断伽辽金方法

• DOI: 10.1016/j.cam.2020.112962
• 发表时间: 2020-12
• 期刊: Journal of Computational and Applied Mathematics
• 影响因子: 2.4
• 作者: Dong, Bo;Wang, Wei
• 通讯作者: Wang, Wei

A New Conservative Discontinuous Galerkin Method via Implicit Penalization for the Generalized Korteweg–de Vries Equation

广义Kortewegâde Vries方程的隐式惩罚的新保守间断伽辽金方法

• DOI: 10.1137/22m1470827
• 发表时间: 2022-12
• 期刊: SIAM Journal on Numerical Analysis
• 影响因子: 2.9
• 作者: Chen, Yanlai;Dong, Bo;Pereira, Rebecca
• 通讯作者: Pereira, Rebecca