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

本计画旨在设计有效的数值格式来模拟复杂的电浆现象。等离子体是一种类似于气体的物质状态，其中一定部分的粒子被电离。 了解等离子体的复杂行为导致了从空间物理、聚变能到高功率微波产生和大规模粒子加速器等领域的重要进展。 有强烈的需要奠定了数学和算法的基础，设计有效的数值方法，使我们能够推进等离子体模拟的基础研究。在这个项目中开发的算法有可能提供高保真度的等离子体物理模拟与可管理的计算成本，并将在聚变装置的多尺度模拟的应用和影响。主要研究员将在专业会议和讲习班上组织特别活动，以促进女性研究人员的参与。该项目为研究生提供了研究培训机会。该项目的目标是在一类称为自适应稀疏网格（aSG）不连续伽辽金（DG）方法的数值方法的设计和分析方面取得重大进展。该方法结合了高阶精度DG求解器，擅长输运模拟和降维技术的aSG方法。该提案的目的是推进时间相关偏微分方程方案的算法基础，并将其推向融合科学多尺度模拟和应用的更广泛竞技场。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，后处理和混合aSG计划的发展。对于一类桥接动力学和流体模型的多尺度动力学问题，通过利用aSG-DG框架提供的多分辨率，研究将利用多尺度模拟工具和分层定义网格上的多分辨率来实现计算加速。该计划将应用于托卡马克装置中失控电子的模拟。该奖项反映了NSF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.