A High Order Adaptive Semi-Lagrangian WENO Method for the Vlasov Equation

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。研究员和她的同事考虑了一种新的高阶自适应半拉格朗日方法，用于动力学等离子体模拟。动力学模拟的主要挑战是巨大的计算成本，从高维（3-D在物理空间和3-D在相空间）和从空间和时间的多尺度特征的问题。为了解决这些问题，本文提出了三种先进的数值方法：1）高阶加权基本无振荡（韦诺）重建的保守半拉格朗日方法; 2）高阶斯特朗分裂谱延迟校正方法，用于桥接不同物种的时间尺度，并在时间上进行高阶精度的积分; 3）网格自适应性，将斯特朗分裂半拉格朗日韦诺方法引入自适应网格加密框架。由于高阶精度和空间和时间的自适应性，研究者和她的合作者希望通过使用相对粗糙和动态自适应的网格，以可承受的计算成本将该算法应用于实际的等离子体应用中。最终目标是实现大规模并行模拟，以预测/确认/解释在广泛应用中观察到的物理现象。研究者和她的合作者旨在开发强大且高效的数值算法。该算法将对等离子体应用产生直接影响，例如开发聚变能，磁层建模等。 此外，还有很大的扩展和应用空间。 该算法可以推广到求解更一般的玻尔兹曼方程。它可以作为一个微观求解器的混合动力学-水动力学模型通过非均匀多尺度方法。 虽然设计的等离子体模拟，所提出的算法可以进一步扩展到天体物理学的应用，半导体器件模拟等。更广泛的影响来自拟议研究的多学科性质。拟议的研究将启动并作为与应用数学家，等离子体物理学家和天体物理学家合作研究工作的坚实基础。合作工作不仅将加快合作双方研究的发展，而且有助于培养具有多样化背景和多学科技能的研究生。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The investigator and her colleagues consider a novel high order adaptive semi-Lagrangian approach for kinetic plasma simulations. The major challenge of kinetic simulations is the huge computational cost, from the high dimensionality (3-D in physical space and 3-D in phasespace) and from the spatial and temporal multi-scale features of the problem. To address these challenges, the proposed methodology consists of three advanced numerical techniques: 1) a conservative semi-Lagrangian method with high order weighted essentially non-oscillatory (WENO) reconstructions; 2) a high order Strang split spectral deferred correction method to bridge time scales of different species and integrate in time with high order accuracy; 3) mesh adaptivity by incorporating the Strang split Semi-Lagrangian WENO method into the framework of adaptive mesh refinement. As a result of high order accuracy and adaptivity in both space and time, the investigator and her collaborators hope to apply the algorithm in realistic plasma applications with affordable computational cost by using relatively coarse and dynamically adaptive mesh. The eventual goal is to enable large-scale parallel simulations in order to predict/confirm/explain the physical phenomenon observed in a broad range of applications.The investigator and her collaborators aim at developing robust and highly efficient numerical algorithms. The well-developed algorithm will have a direct impact in plasma applications, such as developing fusion energy, the modeling of magnetosphere, among many others. Besides, there is large room for further extensions and applications. The algorithm can be extended to solve the more general Boltzmann equation. It can served as a microscopic solver in a mix kinetic-hydrodynamic model via the heterogeneous multi-scale method. While designed for the plasma simulations, the proposed algorithm can be further extended to astrophysics applications, semi-conductor device simulations among many others. The broader impact comes from the multi-disciplinary nature of the proposed research. The proposed research will initiate and serve as a solid foundation for collaborative research work with applied mathematicians, plasma physicists and astrophysicists. The collaborative work will not only expedite the development of the research in both sides of collaborations, but also help training graduate students with a diverse background and multidisciplinary skills.