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

本计画的目标是发展和分析新的间断伽辽金方法来求解不同应用领域的偏微分方程。DG方法是一类使用完全不连续的分段多项式空间作为数值解和检验函数的有限元方法。这些方法鲁棒、紧凑、局部保守，可以处理任意非结构网格，是hp自适应策略的理想选择。该计划的良好性能要求进一步的研究领域，传统上没有解决DG方法。在这项拨款建议中，PI计划在以下方向进行研究：（1）用于求解动力学方程（包括Boltzmann方程和Vlasov方程）的保正DG方法，（2）所提出的方法在太阳能电池/半导体器件模拟和等离子体物理中的应用，（3）一种新的Hamilton-Jacobi方程DG求解器及其在状态约束控制问题中的应用。 拟议的活动之间的算法开发，分析和应用。 为动力学模型和控制问题开发鲁棒的、高阶精度的、具有成本效益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，而且还因为需要对底层物理的深入理解。最终的目标是产生求解器，计算效率高，适合应用程序的需要。PI的工作源于真实的世界应用的计算需求。在这个提议中开发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中具有直接的应用和影响。PI积极与数学，物理，电气工程和化学系的学生和教师互动。 此外，PI将把该项目与研究生的培训相结合，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.