Discontinuous Galerkin Schemes for Fluid, Kinetic, and Multiscale Fluid/Kinetic Models in Plasma Physics Applications

等离子体物理学是研究电离气体动力学的学科。由于等离子体，通常被称为物质的第四种状态，是宇宙中最丰富的普通物质形式，因此存在许多重要的应用问题，理解这些问题将是有利的。这项研究的激励应用是利用激光-等离子体相互作用的高能粒子加速。高能粒子束的产生在产生同步辐射的过程中很重要，同步辐射用于许多应用领域的小尺度成像，包括材料科学，生物学和医学。激光等离子体加速是一种通过用特定调谐的超短激光束撞击等离子体来产生高能电子的机制。这种方法提供了用比标准粒子加速器小得多的设备创造高能粒子的可能性。这种方法的最大挑战是能够产生具有小能量扩散的相对论电子束。本研究的目的是开发高精度和高效率的计算方法来模拟高能电子的加速。这项研究旨在为所谓的等离子体“动力学模型”开发新的计算技术，通过精确地保存某些重要特性（例如，质量和能量守恒以及解的正性）。这些方法将在计算机代码中实现，这些代码将利用现代计算机架构。本研究的主要目的是发展精确有效的计算方法来求解用于模拟等离子体动力学的非线性偏微分方程组。 在这项工作中考虑的数学模型产生的一个重要的应用问题：激光等离子体加速器，它提供了一个有前途的机制，产生高能量的电子击中等离子体与特定调谐的超短激光束。研究将集中在发展有效的高阶间断Galerkin格式的动力Vlasov系统。将解决的两个主要挑战是（1）有效的时间步进程序的发展，允许由物理上重要的时间尺度和（2）高阶离散化，数值上保存重要的物理属性，如质量，动量和能量，以及属性，如积极的解决方案。一种方法，将在这项研究中探索，以提高效率的动力学Vlasov数值方法是耦合动力学Vlasov求解器的流体求解器，以获得多尺度流体/动力学方法。部分工作将与实验物理学家Malte Kaluza博士（耶拿）合作完成，他将帮助指导数学建模，以保持模型与激光等离子体加速器的重要物理学相关。相应地，由此产生的计算结果将被用来影响激光等离子体实验。该研究项目将产生的软件，沿着适当的文档，将作为开放源码软件包DoGPack的一部分在网上免费提供。

Plasma physics is the study of the dynamics of ionized gases. Because plasma, often referred to as the fourth state of matter, is the most abundant form of ordinary matter in the universe, there exist many important application problems for which an understanding would be advantageous. The motivating application for this research is high-energy particle acceleration using laser-plasma interactions. The production of high-energy particle beams is important in the process of creating synchrotron radiation, which is used for small scale imaging in many application areas including materials science, biology, and medicine. Laser-plasma acceleration is a mechanism for generating high-energy electrons by hitting a plasma with a specifically-tuned ultra-short laser beam. This approach provides the possibility to create high-energy particles with a device much smaller than standard particle accelerators. The big challenge with this approach is to be able to generate a relativistic electron beam with a small energy spread. The objective of this research is to develop highly accurate and efficient computational methods for simulating the acceleration of high-energy electrons. This research aims to develop novel computational techniques for so-called 'kinetic models' of plasma that achieve accuracy and robustness by exactly conserving certain important properties (e.g., mass and energy conservation and positivity of the solution). These methods will be implemented in computer code that will take advantage of modern computer architectures. The resulting methods will be used to simulate various scenarios of laser-plasma interactions with the aim to elucidate the mechanisms of electron acceleration and the features of the produced electron bream.The primary objective of this research is to develop accurate and efficient computational methods for solving nonlinear partial differential equations used to model plasma dynamics. The mathematical models considered in this work arise from an important application problem: laser plasma accelerators, which provide a promising mechanism for generating high-energy electrons by hitting a plasma with a specifically-tuned ultra-short laser beam. The research will focus on developing efficient high-order discontinuous Galerkin schemes for kinetic Vlasov systems. The two main challenges that will be addressed are the development of (1) efficient time-stepping procedures that allow for time-steps that are dictated by physically important time scales and (2) high-order discretizations that numerically conserve important physical properties such as mass, momentum, and energy, as well as properties such as the positivity of the solution. One approach that will be explored in this research in order to improve the efficiency of the kinetic Vlasov numerical methods is the coupling of kinetic Vlasov solvers to fluid solvers to obtain a multiscale fluid/kinetic method. Part of the work will be done in collaboration with an experimental physicists, Prof. Dr. Malte Kaluza (Jena), who will help guide the mathematical modeling in order to keep the models relevant to the important physics of laser-plasma accelerators. Correspondingly, the resulting computational results will be used to influence laser-plasma experiments. The software that will result from this research project, along with properdocumentation, will be made freely available on the web as part of the open-source package DoGPack.