Space-time DG-FEMs for Fluid and Kinetic Plasma Models

研究者将与他的学生和合作者一起开发高效的高阶时间步进方法，该方法将与不连续伽辽金空间离散相结合。特别是，将开发自适应时空方法，这将允许(1)明确的局部时间步进，允许在不同的流动状态下不同的时间步进，或者(2)隐式时间步进，这在某些应用中有时是必需的。将集成到这些数值方案中的一个关键特征是以自适应网格细化的形式进行误差控制。我们将研究几种误差估计器，以及几种冲击捕获策略。所得到的数值格式将应用于等离子体物理学中出现的各种模型方程，包括理想磁流体动力学、双流体欧拉-麦克斯韦和动力学Vlasov方程。我们感兴趣的具体应用问题包括太阳日冕环的动力学，天体物理射流的形成和传播，以及无碰撞磁重联的模拟。等离子体是物质的第四种状态（继固体、液体和气体之后），它可以被描述为电离气体（即能够导电的气体）。等离子体有广泛的应用，包括天体物理学和空间物理学，以及在实验室环境中，如磁约束聚变。模拟和理解等离子体中的基本现象一直是科学计算的主题，但许多问题对于现代并行计算机来说仍然过于数字密集。主要的困难是等离子体跨越了广泛的空间和时间尺度。这项研究的范围是发展准确和有效的计算方法，可以更好地解决各种方程，模拟等离子体的行为。本研究的一个关键方面是自适应数值方法的发展，这种方法能够动态估计和控制计算过程中产生的误差。

The investigator, along with his students and collaborators, will develop efficient high-order time-stepping methods that will be used in conjunction with discontinuous Galerkin spatial discretizations. In particular, adaptive space-time methods will be developed that will allow for either (1) explicit local time-stepping that allows for different time-steps in different flow regimes, or, (2) implicit time-stepping, which is sometimes required in certain applications. A key feature that will be integrated into these numerical schemes is error control in the form of adaptive mesh refinement. Several error estimators will be investigated, as well as several shock-capturing strategies. The resulting numerical schemes will be applied to a variety of model equations that arise in plasma physics, including ideal magnetohydrodynamics, two-fluid Euler-Maxwell, and kinetic Vlasov equations. Specific application problems that are of interest include the dynamics of solar coronal loops, the formation and propagation of astrophysical jets, and the simulation of collisionless magnetic reconnection.Plasma is the fourth state of matter (after solid, liquid, and gas), which can be characterized as an ionized gas (i.e., a gas that is able to conduct electricity). Plasma appears in a wide range of applications including astrophysics and space physics, as well as in laboratory settings such as in magnetically confined fusion. Modeling and understanding the basic phenomenon in plasma have long been topics in scientific computing, yet many problems remain far too numerically intensive for modern parallel computers. The main difficulty is that plasmas span a wide range of spatial and temporal scales. The scope of this research is to develop accurate and efficient computational methods that can better solve various equations that model plasma behavior. A key aspect of this research is the development of adaptive numerical methods that are able to dynamically estimate and control the errors that are produced during the course of a computation.