Development of Adaptive Sparse Grid Discontinuous Galerkin Methods for Multiscale Kinetic Simulations in Plasmas

本项目旨在设计模拟复杂等离子体现象的有效数值格式。等离子体是一种类似气体的物质状态，其中一定部分的粒子被电离。对等离子体复杂行为的理解已经在空间物理、核聚变能、高功率微波产生和大规模粒子加速器等领域取得了重要进展。迫切需要为设计有效的数值方法奠定数学和算法基础，以便我们能够推进等离子体模拟的基础研究。本项目开发的算法具有在等离子体物理中提供高保真度模拟的潜力，计算成本可控，并将在聚变装置的多尺度模拟中具有应用和影响。首席研究员（PI）将在专业会议和研讨会上组织特别活动，以促进女性研究人员的参与。本项目为研究生提供研究训练机会。该项目的目标是在设计和分析一类称为自适应稀疏网格（aSG）不连续伽辽金（DG）方法的数值方法方面取得重大进展。该方法结合了高阶精确DG求解和aSG方法的降维技术。本提案的目的是推进时间相关偏微分方程方案的算法基础，并将其推向更广泛的多尺度模拟和融合科学应用领域。PI将研究几个基本问题，包括CFL条件的分析，多尺度时间步进的发展，后处理和混合aSG方案。对于一类连接动力学模型和流体模型的多尺度动力学问题，利用aSG-DG框架提供的多分辨率，研究将利用多尺度仿真工具和分层定义网格的多分辨率来实现计算加速。这些方案将应用于托卡马克装置中失控电子的模拟。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project aims at designing efficient numerical schemes for simulating complex plasma phenomena. Plasma is a state of matter similar to gas in which a certain portion of the particles is ionized. Understanding the complex behaviors of plasmas has led to important advances in areas ranging from space physics, fusion energy, to high-power microwave generation and large scale particle accelerators. There is strong need for laying out mathematical and algorithmic foundations for the design of efficient numerical methods so that we can advance basic research in plasma simulations. The algorithms developed in this project have the potential to provide high fidelity simulations in plasma physics with manageable computational cost and will have applications and impacts in multiscale simulations in fusion devices. The principal investigator (PI) will organize special events at professional meetings and workshops to promote the participation of female researchers. This project provides research training opportunities for graduate students. The objective of the project is to make significant advances on the design and analysis of a class of numerical methods called adaptive sparse grid (aSG) discontinuous Galerkin (DG) methods. The methods incorporate high order accurate DG solver that excels at transport simulations and the dimension reduction technique by aSG approach. The aim of this proposal is to advance the algorithmic foundations of the schemes for time-dependent PDEs, and push them onto the broader arena of multiscale simulations and applications for fusion science. The PI will investigate several fundamental issues including the analysis of CFL conditions, development of multiscale time stepping, postprocessing and hybrid aSG schemes. For a class of multiscale kinetic problems bridging kinetic and fluid models, by utilizing the multiresolution offered in the aSG-DG framework, the research will take advantage of both multiscale simulation tools and multiresolution on hierarchically defined meshes to achieve acceleration in computations. The schemes will be applied to simulations of runaway electrons in tokamak devices.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.