在本文中，我们提出了一种参数化最大原理保持（MPP）通量限制器，最初在[Z. Xu, Math. Comp., (2013), in press]中开发，用于求解Vlasov方程的半拉格朗日有限差分加权本质非振荡方案。MPP通量限制器被证明在无任何时间步限制的情况下，能够保持半拉格朗日有限差分方案的四阶精度。对Vlasov-Poisson系统的数值研究展示了所提出方法的性能及其在保持概率分布函数非负性的同时维持高阶精度的能力。
英文摘要：
In this paper, we propose the parametrized maximum principle preserving (MPP) flux limiter, originally developed in [Z. Xu, Math. Comp., (2013), in press], to the semi- Lagrangian finite difference weighted essentially non-oscillatory scheme for solving the Vlasov equation. The MPP flux limiter is proved to maintain up to fourth order accuracy for the semi-Lagrangian finite difference scheme without any time step restriction. Numerical studies on the Vlasov-Poisson system demonstrate the performance of the proposed method and its ability in preserving the positivity of the probability distribution function while maintaining the high order accuracy.
This paper develops the structure-preserving finite volume weighted essentially non-oscillatory (WENO) hybrid schemes for the shallow water equations under the arbitrary Lagrangian-Eulerian (ALE) framework, dubbed as ALE-WENO schemes. The WENO hybrid reconstruction is adopted on moving meshes, which distinguishes the smooth, non-smooth, and transition stencils by a simple smoothness detector. To maintain the positivity preserving and the well-balanced properties of the ALE-WENO schemes, we adapt the positivity preserving limiter and the well-balanced approaches on static meshes to moving meshe…
The weighted essentially non-oscillatory (WENO) schemes are a popular class of high order accurate numerical methods for solving hyperbolic partial differential equations (PDEs). However when the spatial dimensions are high, the number of spatial grid points increases significantly. It leads to large amount of operations and computational costs in the numerical simulations by using nonlinear high order accuracy WENO schemes such as a fifth order WENO scheme. How to achieve fast simulations by high order WENO methods for high spatial dimension hyperbolic PDEs is a challenging and important ques…
We present fourth-order conservative non-splitting semi-Lagrangian (SL) Hermite essentially non-oscillatory (HWENO) schemes for linear transport equations with applications for nonlinear problems including the Vlasov-Poisson system, the guiding center Vlasov model, and the incompressible Euler equations in the vorticity-stream function formulation. The proposed SL HWENO schemes combine a weak formulation of the characteristic Galerkin method with two newly constructed HWENO reconstruction methods. Fourth-order accuracy is accomplished in both space and time under a non-splitting setting. Mass …
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）
The semi-Lagrangian discontinuous Galerkin method, coupled with a splitting approach in time, has recently been introduced for the Vlasov--Poisson equation. Since these methods are conservative, local in space, and able to limit numerical diffusion, they are considered a promising alternative to more traditional semi-Lagrangian schemes. In this paper we study the conservation of important physical invariants and the long time behavior of the semi-Lagrangian discontinuous Galerkin method. To that end we conduct a theoretical analysis and perform a number of numerical simulations. In particular,…
Arbitrary-order time-accurate semi-Lagrangian spectral approximations of the Vlasov-Poisson system
Università degli Studi di Camerino（卡梅里诺大学）；Università degli Studi di Modena e Reggio Emilia（摩德纳和雷焦艾米利亚大学）；Istituto di Matematica Applicata e Tecnologie Informatiche, Consiglio Nazionale delle Ricerche（意大利国家研究委员会应用数学与信息技术研究所）；Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）
The purpose of the present paper is to compare two semi-Lagrangian methods in the context of the four-dimensional Vlasov--Poisson equation. More specifically, our goal is to compare the performance of the more recently developed semi-Lagrangian discontinuous Galerkin scheme with the de facto standard in Eulerian Vlasov simulation (i.e. using cubic spline interpolation). To that end, we perform simulations for nonlinear Landau damping and a two-stream instability and provide benchmarks for the SeLaLib and sldg codes (both on a workstation and using MPI on a cluster). We find that the semi-Lagra…
A Semi-Lagrangian Spectral Method for the Vlasov-Poisson System based on Fourier, Legendre and Hermite Polynomials
Dipartimento di Matematica, Università degli Studi di Camerino（卡梅里诺大学数学系）；Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Università degli Studi di Modena e Reggio Emilia（摩德纳和雷焦艾米利亚大学物理、信息和数学科学系）；Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）
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 Hybrid semi-Lagrangian Flow Mapping Approach for Vlasov Systems: Combining Iterative and Compositional Flow Maps
IRFM - CEA Cadarache（法国原子能和替代能源委员会卡达拉舍聚变研究设施）；Centre for mathematical Plasma Astrophysics, Department of Mathematics, KU Leuven（荷语鲁汶大学数学系数学等离子体天体物理学中心）；Royal Belgian Institute for Space Aeronomy, Solar-Terrestrial Centre of Excellence（比利时皇家空间气象研究所太阳-地球卓越中心）；McGill University（麦吉尔大学）
We propose a new Eulerian-Lagrangian Runge-Kutta finite volume method for numerically solving convection and convection-diffusion equations. Eulerian-Lagrangian and semi-Lagrangian methods have grown in popularity mostly due to their ability to allow large time steps. Our proposed scheme is formulated by integrating the PDE on a space-time region partitioned by approximations of the characteristics determined from the Rankine-Hugoniot jump condition; and then rewriting the time-integral form into a time differential form to allow application of Runge-Kutta (RK) methods via the method-of-lines …
In this paper, we propose and analyse a reconstruction technique which enables one to design high-order conservative semi-Lagrangian schemes for kinetic equations. The proposed reconstruction can be obtained by taking the sliding average of a given polynomial reconstruction of the numerical solution. A compact representation of the high order conservative reconstruction in one and two space dimension is provided, and its mathematical properties are analyzed. To demonstrate the performance of proposed technique, we consider implicit semi-Lagrangian schemes for kinetic-like equations such as the…
The Arnold-Beltrami-Childress (ABC) flow and the Kolmogorov flow are three dimensional periodic divergence free velocity fields that exhibit chaotic streamlines. We are interested in front speed enhancement in G-equation of turbulent combustion by large intensity ABC and Kolmogorov flows. We give a quantitative construction of the ballistic orbits of ABC and Kolmogorov flows, namely those with maximal large time asymptotic speeds in a coordinate direction. Thanks to the optimal control theory of G-equation (a convex but non-coercive Hamilton-Jacobi equation), the ballistic orbits serve as admi…
