Multiscale and Hybridizable Discontinuous Galerkin Methods for Dispersive Equations and Systems

本项目致力于开发新的计算方法，有效地解决色散方程和系统，包括时间无关的薛定谔方程和Korteweg-de弗里斯（KdV）型方程在多维空间和系统。 薛定谔方程在量子力学系统的研究中起着核心作用，并广泛用于纳米半导体器件中量子输运的模拟。提出的薛定谔方程的多尺度方法将在量子力学的研究中产生积极的影响，并在超快，低功耗和高功能的纳米半导体器件的应用中具有巨大的潜力。KdV型方程和方程组在流体力学、非线性光学、声学、等离子体物理、玻色-爱因斯坦凝聚等领域有着广泛的应用。本文所提出的KdV型方程和系统的数值模拟方法将有助于理解理论上尚未解决的问题，并为实际应用中的非线性波浪模拟提供准确、高效的数值工具。 本论文的主要研究内容包括：（1）一维、系统和二维Schrodinger方程的多尺度间断Galerkin（DG）方法的发展和分析，以及在粗网格上模拟纳米半导体结构的方法;（2）求解多维KdV型方程和KdV型系统的可杂交间断Galerkin（HDG）方法的设计和误差分析，（3）高效求解KdV型非线性方程组的IMEX HDG-DG格式的设计。为了在粗网格上有效地求解薛定谔方程的高振荡解，多尺度DG方法将解的振荡性质和多尺度结合到非多项式基函数中。对于多维系统中的KdV型方程，PI将设计新的HDG方法，研究其收敛性和守恒性，并将其与IMEX格式中的其他DG方法相结合，以获得时间和空间上的高阶解，避免时间过小，该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准。

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.