Development of Discontinuous Galerkin Methods for Kinetic Transport Models and Control Problems with State Constraints

本项目的目的是发展和分析新的不连续伽辽金（DG）方法来求解不同应用领域的偏微分方程。DG法是一类利用完全不连续分段多项式空间求解数值解和测试函数的有限元方法。这些鲁棒、紧凑、局部保守的方法可以处理任意非结构化网格，是hp自适应策略的理想选择。该方案的良好性能要求在传统的DG方法无法解决的领域进行进一步的研究。在此资助申请中，PI计划在以下方向进行研究：(1)求解动力学方程的保正DG方法，包括Boltzmann方程和Vlasov方程；(2)将所提出的方法应用于太阳能电池/半导体器件模拟和等离子体物理；(3)Hamilton-Jacobi方程的新型DG求解器及其在状态约束控制问题中的应用。提议的活动介于算法开发、分析和应用之间。为动力学模型和控制问题开发鲁棒、高阶精确、经济高效的数值算法是非常具有挑战性的，这不仅是因为这些模型的高维性，还因为需要对底层物理有深刻的理解。最终目标是产生计算效率高且适合应用程序需要的求解器。PI的工作源于现实世界应用程序的计算需求。本提案中提出的许多想法将在半导体器件模拟、高效燃料电池建模、控制问题和等离子体物理方面有直接的应用和影响。PI积极与数学、物理、电子工程和化学系的学生和教师互动。此外，项目负责人将把该项目与研究生的培训结合起来，以便在更广泛的背景下进行交流。

The objective of this project is to develop and analyze novel discontinuous Galerkin (DG) methods for solving partial differential equations arising from various application areas. The DG method is a class of finite element methods using completely discontinuous piecewise polynomial space for the numerical solution and the test functions. Those robust, compact, locally conservative methods can treat arbitrarily unstructured meshes and are ideal for hp-adaptive strategies. The good properties of the scheme call for further research in areas that are traditionally not solved by DG methods. In this grant proposal, the PI plans to conduct research in the following directions: (1) a positivity-preserving DG method for solving the kinetic equations, including the Boltzmann equations and Vlasov equations, (2) application of the proposed method to solar cell/semiconductor device simulations and plasma physics, (3) a novel DG solver for the Hamilton-Jacobi equations and its applications in control problems with state constraints. The proposed activity lies between algorithm development, analysis and applications. Developing robust, high-order accurate, cost-efficient numerical algorithms for kinetic models and control problem is very challenging, not only because of the high dimensionality of such models, but also because of the fact that a deep understanding of the underlying physics is required. The eventual goal is to produce solvers that are computationally efficient and suit the need for applications. The PI's work arises from the computational demand of real world applications. Many ideas developed in this proposal will have straightforward applications and impacts in semiconductor device simulations, high-efficiency fuel cell modeling, control problems and plasma physics. The PI actively interacts with students and faculty members in mathematics, physics, electrical engineering and chemistry departments. In addition, the PI will integrate the project with the training of graduate students in order to communicate in a broader context.