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。研究人员和她的同事们考虑了一种新的高阶自适应半拉格朗日方法来模拟动力学等离子体。动力学模拟的主要挑战是来自高维(物理空间为3维，相空间为3维)以及问题的时空多尺度特征的巨大计算成本。为了应对这些挑战，该方法包括三个先进的数值技术：1)具有高阶加权基本无振荡(WENO)重构的保守半拉格朗日方法；2)连接不同物种时间尺度并以高精度进行时间积分的高阶Strang分裂谱延迟修正方法；3)通过将Strang Split半拉格朗日WENO方法引入自适应网格加密框架来实现网格自适应。由于该算法在空间和时间上的高精度和自适应性，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，以负担得起的计算成本将该算法应用于现实的等离子体应用中。最终目标是实现大规模并行模拟，以便预测/确认/解释在广泛应用中观察到的物理现象。研究人员和她的合作者致力于开发健壮和高效的数值算法。完善的算法将对等离子体应用产生直接影响，如开发聚变能量、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Boltzmann方程。通过非均相多尺度方法，它可以作为混合动力学-流体动力学模型中的微观求解器。虽然该算法是为等离子体模拟而设计的，但它可以进一步扩展到天体物理应用、半导体器件模拟等许多领域。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为与应用数学家、等离子体物理学家和天体物理学家的合作研究工作奠定坚实的基础。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不同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.