Vlasov-Poisson 方程描述无碰撞等离子体的演化，通过概率密度函数（PDF）表示，该函数通过静电力自相互作用。数值求解该系统的主要困难之一是与中等至大速度相关的 PDF 部分带来的严格时间步长限制。等离子体物理社区中消除这些时间步长限制的主流方法是所谓的粒子网格（PIC）方法，它将分布函数离散为一组宏粒子，而电场在网格上表示。存在几种替代方法，包括完全拉格朗日、完全欧拉和所谓的半拉格朗日方法。本工作的重点是半拉格朗日方法，它从 PDF 和电场的基于网格的欧拉表示开始，然后通过拉格朗日动力学演化 PDF，最后将演化后的场投影回原始欧拉网格。特别地，我们在本工作中开发了一种方法，通过相空间中的高阶间断 Galerkin（DG）方法和时间上的算子分裂半拉格朗日方法离散 1+1 维 Vlasov-Poisson 系统。通过 Strang 算子分裂相对容易实现二阶时间精度。通过额外的工作，使用高阶分裂和高阶特征线法，我们还展示了如何将该格式推进到四阶时间精度。我们展示了如何解析所有拉格朗日动力学，使得质量精确守恒、保持正性并实现高阶精度。泊松方程通过最小模板局部间断 Galerkin（LDG）方法求解至高阶。我们在几个标准测试案例上测试了所提出的格式。
英文摘要：
The Vlasov-Poisson equations describe the evolution of a collisionless plasma, represented through a probability density function (PDF) that self-interacts via an electrostatic force. One of the main difficulties in numerically solving this system is the severe time-step restriction that arises from parts of the PDF associated with moderate-to-large velocities. The dominant approach in the plasma physics community for removing these time-step restrictions is the so-called particle-in-cell (PIC) method, which discretizes the distribution function into a set of macro-particles, while the electric field is represented on a mesh. Several alternatives to this approach exist, including fully Lagrangian, fully Eulerian, and so-called semi-Lagrangian methods. The focus of this work is the semi-Lagrangian approach, which begins with a grid-based Eulerian representation of both the PDF and the electric field, then evolves the PDF via Lagrangian dynamics, and finally projects this evolved field back onto the original Eulerian mesh. In particular, we develop in this work a method that discretizes the 1+1 Vlasov-Poisson system via a high-order discontinuous Galerkin (DG) method in phase space, and an operator split, semi-Lagrangian method in time. Second-order accuracy in time is relatively easy to achieve via Strang operator splitting. With additional work, using higher-order splitting and a higher-order method of characteristics, we also demonstrate how to push this scheme to fourth-order accuracy in time. We show how to resolve all of the Lagrangian dynamics in such a way that mass is exactly conserved, positivity is maintained, and high-order accuracy is achieved. The Poisson equation is solved to high-order via the smallest stencil local discontinuous Galerkin (LDG) approach. We test the proposed scheme on several standard test cases.
We prove the convergence of discontinuous Galerkin approximations for the Vlasov-Poisson system written as an hyperbolic system using Hermite polynomials in velocity. To obtain stability properties, we introduce a suitable weighted L 2 space, with a time dependent weight, and first prove global stability for the weighted L 2 norm and propagation of regularity. Then we prove error estimates between the numerical solution and the smooth solution to the Vlasov-Poisson system.
We study a class of spatial discretizations for the Vlasov-Poisson system written as an hyperbolic system using Hermite polynomials. In particular, we focus on spectral methods and discontinuous Galerkin approximations. To obtain L 2 stability properties, we introduce a new L 2 weighted space, with a time dependent weight. For the Hermite spectral form of the Vlasov-Poisson system, we prove conservation of mass, momentum and total energy, as well as global stability for the weighted L 2 norm. These properties are then discussed for several spatial discretizations. Finally, numerical simulation…
Discrete hypocoercive estimates for discontinuous Galerkin methods: application to the Vlasov-Poisson-Fokker-Planck system
Xiamen University（厦门大学）；Centre Inria de l’Université de Lille（里尔大学Inria中心）；University of Science and Technology of China（中国科学技术大学）；Institut de Mathématiques de Toulouse, Université de Toulouse（图卢兹数学研究所，图卢兹大学）
High-order entropy-stable discontinuous Galerkin methods for the compressible Euler and Navier-Stokes equations require the positivity of thermodynamic quantities in order to guarantee their well-posedness. In this work, we introduce a positivity limiting strategy for entropy-stable discontinuous Galerkin discretizations constructed by blending high order solutions with a low order positivity-preserving discretization. The proposed low order discretization is semi-discretely entropy stable, and the proposed limiting strategy is positivity preserving for the compressible Euler and Navier-Stokes…
This paper considers the discontinuous Galerkin (DG) methods for solving the Vlasov-Maxwell (VM) system, a fundamental model for collisionless magnetized plasma. The DG methods provide accurate numerical description with conservation and stability properties. However, to resolve the high dimensional probability distribution function, the computational cost is the main bottleneck even for modern-day supercomputers. This work studies the applicability of a post-processing technique to the DG solution to enhance its accuracy and resolution for the VM system. In particular, we prove the superconve…
A high order semi-Lagrangian discontinuous Galerkin method for the two-dimensional incompressible Euler equations and the guiding center Vlasov model without operator splitting
University of Delaware（特拉华大学）；Texas Tech University（德克萨斯理工大学）
High order schemes are known to be unstable in the presence of shock discontinuities or under-resolved solution features, and have traditionally required additional filtering, limiting, or artificial viscosity to avoid solution blow up. Entropy stable schemes address this instability by ensuring that physically relevant solutions satisfy a semi-discrete entropy inequality independently of discretization parameters. However, additional measures must be taken to ensure that solutions satisfy physical constraints such as positivity. In this work, we present a high order entropy stable discontinuo…
In this paper, we propose a semi-Lagrangian discontinuous Galerkin method coupled with Runge-Kutta exponential integrators (SLDG-RKEI) for nonlinear Vlasov dynamics. The commutator-free Runge-Kutta (RK) exponential integrators (EI) were proposed by Celledoni, et al. (FGCS, 2003). In the nonlinear transport setting, the RKEI can be used to decompose the evolution of the nonlinear transport into a composition of a sequence of linearized dynamics. The resulting linearized transport equations can be solved by the semi-Lagrangian (SL) discontinuous Galerkin (DG) method proposed in Cai, et al. (JSC,…
A discontinuous Galerkin method for the Vlasov-Poisson system
Applied Research Laboratories, University of Texas at Austin（德克萨斯大学奥斯汀分校应用研究实验室）；GE Global Research（通用电气全球研发中心）；Department of Mathematics and ICES, University of Texas at Austin（德克萨斯大学奥斯汀分校数学系与计算科学与工程研究所）；Department of Physics and Institute for Fusion Studies, University of Texas at Austin（德克萨斯大学奥斯汀分校物理系与聚变研究所）；Computing Laboratory, The University of Oxford（牛津大学计算实验室）
A well-balanced positivity-preserving discontinuous Galerkin method for shallow water models with variable density
Chongqing University（重庆大学）；College of Mathematics and Statistics, Chongqing University（重庆大学数学与统计学院）；School of arts and sciences, Chengdu College of University of Electronic Science and Technology of China（电子科技大学成都学院文理学院）；Sichuan International Studies University（四川外国语大学）
Positivity-preserving and entropy-bounded discontinuous Galerkin method for the chemically reacting, compressible Euler equations. Part II: The multidimensional case
In this paper, we propose a novel Local Macroscopic Conservative (LoMaC) low rank tensor method with discontinuous Galerkin (DG) discretization for the physical and phase spaces for simulating the Vlasov-Poisson (VP) system. The LoMaC property refers to the exact local conservation of macroscopic mass, momentum and energy at the discrete level. The recently developed LoMaC low rank tensor algorithm (arXiv:2207.00518) simultaneously evolves the macroscopic conservation laws of mass, momentum and energy using the kinetic flux vector splitting; then the LoMaC property is realized by projecting th…
High order entropy stable schemes provide improved robustness for computational simulations of fluid flows. However, additional stabilization and positivity preserving limiting can still be required for variable-density flows with under-resolved features. We demonstrate numerically that entropy stable DG methods which incorporate an "entropy projection" are less likely to require additional limiting to retain positivity for certain types of flows. We conclude by investigating potential explanations for this observed improvement in robustness.
We present a novel class of locally conservative, entropy stable and well-balanced discontinuous Galerkin (DG) methods for the nonlinear shallow water equation with a non-flat bottom topography. The major novelty of our work is the use of velocity field as an independent solution unknown in the DG scheme, which is closely related to the entropy variable approach to entropy stable schemes for system of conservation laws proposed by Tadmor [22] back in 1986, where recall that velocity is part of the entropy variable for the shallow water equations. Due to the use of velocity as an independent so…
In this paper, we present a positivity-preserving limiter for nodal Discontinuous Galerkin disctretizations of the compressible Euler equations. We use a Legendre-Gauss-Lobatto (LGL) Discontinuous Galerkin Spectral Element Method (DGSEM) and blend it locally with a consistent LGL-subcell Finite Volume (FV) discretization using a hybrid FV/DGSEM scheme that was recently proposed for entropy stable shock capturing. We show that our strategy is able to ensure robust simulations with positive density and pressure when using the standard and the split-form DGSEM. Furthermore, we show the applicabil…
Energy-, momentum-, density-, and positivity-preserving spatio-temporal discretizations for the nonlinear Landau collision operator with exact H-theorems
Princeton Plasma Physics Laboratory（普林斯顿等离子体物理实验室）；Courant Institute of Mathematical Sciences（库朗数学科学研究所）；Max-Planck-Institut für Plasmaphysik（马克斯·普朗克等离子体物理研究所）；Technische Universität München（慕尼黑工业大学）
