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

这个项目致力于开发新的计算方法来有效地解决具有守恒的物理性质或高度振荡的波解的问题。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进行准确和稳定的模拟。它们将在流体动力学、非线性光学、等离子体物理和玻色-爱因斯坦凝聚等领域中得到应用。新的多尺度方法能够准确、高效地捕捉到高度振荡的波解。它们将对量子力学的研究产生积极的影响，并在设计超快、低功耗的纳米电子器件方面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。该项目包括以下主题：(1)深入研究Korteweg-de Vries(KdV)方程的隐式罚参数守恒间断Galerkin(DG)方法，(2)通过隐式惩罚发展更复杂的具有守恒性质的波动模型的守恒DG方法，包括Hirota-SatSuma耦合KdV系统、Schrodinger-KdV系统、ABCD-Boussinesq系统和二维Zakharov-Kuznetsov(ZK)方程和Kadomtsev-Petviashvili(KP)方程，(3)设计、分析、研究以及基于多尺度基的杂交间断伽辽金(HDG)方法的实现，用于在粗网格上有效地捕捉薛定谔方程的高振荡解。前两个主题中的新想法是通过隐式惩罚来加强守恒性，这可以推广到以物理量守恒为特征的其他类型的问题。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。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.