球坐标动量空间下相对论Vlasov方程的数值算法

针对相对论Vlasov方程动量区间跨度大、难以计算的困难,将相对论Vlasov方程在球坐标动量空间中进行数值求解.对相对论Vlasov方程球坐标动量空间构造4阶非分裂守恒型数值格式.数值模拟相对论Landau阻尼问题并与解析理论进行比较,验证数值模型和算法的有效性.对激光等离子体相互作用进行初步模拟分析,表明通过采用球坐标下的动量空间,可在相对较少动量网格情形下,获得与粒子模拟可相互验证的结果.     

Exploring the thermodynamic limit of Hamiltonian models: convergence to the Vlasov equation

We here discuss the emergence of quasistationary states (QSS), a universal feature of systems with long-range interactions. With reference to the Hamiltonian mean-field model, numerical simulations are performed based on both the original N-body setting and the continuum Vlasov model which is supposed to hold in the thermodynamic limit. A detailed comparison unambiguously demonstrates that the Vlasov-wave system provides the correct framework to address the study of QSS. Further, analytical calculations based on Lynden-Bell's theory of violent relaxation are shown to result in accurate predictions. Finally, in specific regions of parameters space, Vlasov numerical solutions are shown to be affected by small scale fluctuations, a finding that points to the need for novel schemes able to account for particle correlations.     

Positivity preserving semi-Lagrangian discontinuous Galerkin formulation: Theoretical analysis and application to the Vlasov–Poisson system

Semi-Lagrangian (SL) methods have been very popular in the Vlasov simulation community , , , , , ,  and . In this paper, we propose a new Strang split SL discontinuous Galerkin (DG) method for solving the Vlasov equation. Specifically, we apply the Strang splitting for the Vlasov equation [6], as a way to decouple the nonlinear Vlasov system into a sequence of 1-D advection equations, each of which has an advection velocity that only depends on coordinates that are transverse to the direction of propagation. To evolve the decoupled linear equations, we propose to couple the SL framework with the semi-discrete DG formulation. The proposed SL DG method is free of time step restriction compared with the Runge–Kutta DG method, which is known to suffer from numerical time step limitation with relatively small CFL numbers according to linear stability analysis. We apply the recently developed positivity preserving (PP) limiter [37], which is a low-cost black box procedure, to our scheme to ensure the positivity of the unknown probability density function without affecting the high order accuracy of the base SL DG scheme. We analyze the stability and accuracy properties of the SL DG scheme by establishing its connection with the direct and weak formulations of the characteristics/Lagrangian Galerkin method [23]. The quality of the proposed method is demonstrated via basic test problems, such as linear advection and rigid body rotation, and via classical plasma problems, such as Landau damping and the two stream instability.     

A numerical scheme for the integration of the Vlasov–Poisson system of equations, in the magnetized case

We present a numerical algorithm for the solution of the Vlasov–Poisson system of equations, in the magnetized case. The numerical integration is performed using the well-known “splitting” method in the electrostatic approximation, coupled with a finite difference upwind scheme; finally the algorithm provides second order accuracy in space and time. The cylindrical geometry is used in the velocity space, in order to describe the rotation of the particles around the direction of the external uniform magnetic field.Using polar coordinates, the integration of the Vlasov equation is very simplified in the velocity space with respect to the cartesian geometry, because the rotation in the velocity cartesian space corresponds to a translation along the azimuthal angle in the cylindrical reference frame. The scheme is intrinsically symplectic and significatively simpler to implement, with respect to a cartesian one. The numerical integration is shown in detail and several conservation tests are presented, in order to control the numerical accuracy of the code and the time evolution of the entropy, strictly related to the filamentation problem for a kinetic model, is discussed.     

A high-order cell-centered Lagrangian scheme for compressible fluid flows in two-dimensional cylindrical geometry

The goal of this paper is to present high-order cell-centered schemes for solving the equations of Lagrangian gas dynamics written in cylindrical geometry. A node-based discretization of the numerical fluxes is obtained through the computation of the time rate of change of the cell volume. It allows to derive finite volume numerical schemes that are compatible with the geometric conservation law (GCL). Two discretizations of the momentum equations are proposed depending on the form of the discrete gradient operator. The first one corresponds to the control volume scheme while the second one corresponds to the so-called area-weighted scheme. Both formulations share the same discretization for the total energy equation. In both schemes, fluxes are computed using the same nodal solver which can be viewed as a two-dimensional extension of an approximate Riemann solver. The control volume scheme is conservative for momentum, total energy and satisfies a local entropy inequality in its first-order semi-discrete form. However, it does not preserve spherical symmetry. On the other hand, the area-weighted scheme is conservative for total energy and preserves spherical symmetry for one-dimensional spherical flow on equi-angular polar grid. The two-dimensional high-order extensions of these two schemes are constructed employing the generalized Riemann problem (GRP) in the acoustic approximation. Many numerical tests are presented in order to assess these new schemes. The results obtained for various representative configurations of one and two-dimensional compressible fluid flows show the robustness and the accuracy of our new schemes.     

An improved high-order scheme for DNS of low Mach number turbulent reacting flows based on stiff chemistry solver

We present an improved numerical scheme for numerical simulations of low Mach number turbulent reacting flows with detailed chemistry and transport. The method is based on a semi-implicit operator-splitting scheme with a stiff solver for integration of the chemical kinetic rates, developed by Knio et al. [O.M. Knio, H.N. Najm, P.S. Wyckoff, A semi-implicit numerical scheme for reacting flow II. Stiff, operator-split formulation, Journal of Computational Physics 154 (2) (1999) 428–467]. Using the material derivative form of continuity equation, we enhance the scheme to allow for large density ratio in the flow field. The scheme is developed for direct numerical simulation of turbulent reacting flow by employing high-order discretization for the spatial terms. The accuracy of the scheme in space and time is verified by examining the grid/time-step dependency on one-dimensional benchmark cases: a freely propagating premixed flame in an open environment and in an enclosure related to spark-ignition engines. The scheme is then examined in simulations of a two-dimensional laminar flame/vortex-pair interaction. Furthermore, we apply the scheme to direct numerical simulation of a homogeneous charge compression ignition (HCCI) process in an enclosure studied previously in the literature. Satisfactory agreement is found in terms of the overall ignition behavior, local reaction zone structures and statistical quantities. Finally, the scheme is used to study the development of intrinsic flame instabilities in a lean H₂/air premixed flame, where it is shown that the spatial and temporary accuracies of numerical schemes can have great impact on the prediction of the sensitive nonlinear evolution process of flame instability.     

A parallel Vlasov solver based on local cubic spline interpolation on patches

A method for computing the numerical solution of Vlasov type equations on massively parallel computers is presented. In contrast with Particle In Cell methods which are known to be noisy, the method is based on a semi-Lagrangian algorithm that approaches the Vlasov equation on a grid of phase space. As this kind of method requires a huge computational effort, the simulations are carried out on parallel machines. To that purpose, we present a local cubic splines interpolation method based on a domain decomposition, e.g. devoted to a processor. Hermite boundary conditions between the domains, using ad hoc reconstruction of the derivatives, provide a good approximation of the global solution. The method is applied on various physical configurations which show the ability of the numerical scheme.     

A comparative study of the performance of high-resolution non-oscillating advection schemes in the context of the motion induced by mixed region in a stratified fluid

The problem of a two-dimensional fully mixed region collapsing in continuously density-stratified medium is considered. This article deals with the numerical treatment of the advective terms in the Navier-Stokes equations in the Oberbeck-Boussinesq approximation. Comparisons are performed between the upwind scheme, flux-limiter schemes, namely, Minmod, Superbee, Van Leer, and Monotonized Centred (MC), the monotone adaptive stencil schemes, namely, ENO3 and SMIF, and weighted stencil scheme WENO5. We used laboratory experimental data of Wu as a benchmark test to compare performance of different numerical approaches. It is found that the flux-limiter schemes have the smallest numerical diffusion. The WENO5 scheme describes more accurately the width of collapse region variation with time. All considered schemes give a realistic pattern of internal gravity waves generated by collapse region.     

Implicit schemes for real-time lattice gauge theory

We develop new gauge-covariant implicit numerical schemes for classical real-time lattice gauge theory. A new semi-implicit scheme is used to cure a numerical instability encountered in three-dimensional classical Yang-Mills simulations of heavy-ion collisions by allowing for wave propagation along one lattice direction free of numerical dispersion. We show that the scheme is gauge covariant and that the Gauss constraint is conserved even for large time steps.     

A Free-Lagrange Augmented Godunov Method for the Simulation of Elastic–Plastic Solids

A Lagrangian finite-volume Godunov scheme is extended to simulate two-dimensional solids in planar geometry. The scheme employs an elastic–perfectly plastic material model, implemented using the method of radial return, and either the ‘stiffened’ gas or Osborne equation of state to describe the material. The problem of mesh entanglement, common to conventional two-dimensional Lagrangian schemes, is avoided by utilising the free-Lagrange Method. The Lagrangian formulation enables features convecting at the local velocity, such as material interfaces, to be resolved with minimal numerical dissipation. The governing equations are split into separate subproblems and solved sequentially in time using a time-operator split procedure. Local Riemann problems are solved using a two-shock approximate Riemann solver, and piecewise-linear data reconstruction is employed using a MUSCL-based approach to improve spatial accuracy. To illustrate the effectiveness of the technique, numerical simulations are presented and compared with results from commercial fixed-connectivity Lagrangian and smooth particle hydrodynamics solvers (AUTODYN-2D). The simulations comprise the low-velocity impact of an aluminium projectile on a semi-infinite target, the collapse of a thick-walled beryllium cylinder, and the high-velocity impact of cylindrical aluminium and steel projectiles on a thin aluminium target. The analytical solution for the collapse of a thick-walled cylinder is also presented for comparison.     

低阶格式在大涡模拟计算中的适用性

采用三维Taylor-Green涡作为研究对象,利用工程中常用的低阶数值格式,研究格式本身的数值误差对大涡模拟计算的影响.结果表明:三种数值格式的数值耗散行为都与亚格子模型行为类似,即在小雷诺数下,流场比较光滑时,耗散很小,当雷诺数增加,流动转捩为湍流,流场梯度增大,耗散显著增大.对于MUSCL格式和二阶有界中心格式,在高雷诺数下,亚格子尺度模型没有明显改善计算结果,但也没有使计算结果恶化.中心格式相比其它两种格式,数值耗散最小,但是在高雷诺数湍流情况下,中心格式的数值耗散仍然主导了能量的耗散,再添加亚格子模型,计算结果反而变得稍差.对于工程中的低阶格式而言,采用中心格式计算大涡模拟是比较好的选择,而且在计算不存在稳定性问题时,采用不添加亚格子模型的隐式大涡模拟效果更好.     

A cell-centered Lagrangian scheme with the preservation of symmetry and conservation properties for compressible fluid flows in two-dimensional cylindrical geometry

We develop a new cell-centered control volume Lagrangian scheme for solving Euler equations of compressible gas dynamics in cylindrical coordinates. The scheme is designed to be able to preserve one-dimensional spherical symmetry in a two-dimensional cylindrical geometry when computed on an equal-angle-zoned initial grid. Unlike many previous area-weighted schemes that possess the spherical symmetry property, our scheme is discretized on the true volume and it can preserve the conservation property for all the conserved variables including density, momentum and total energy. Several two-dimensional numerical examples in cylindrical coordinates are presented to demonstrate the performance of the scheme in terms of symmetry, accuracy and non-oscillatory properties.     

Nonlinear ultrasonic standing waves: two-dimensional simulations in bubbly liquids

We present the results of numerical predictions for analyzing the behavior of nonlinear ultrasonic standing waves in two-dimensional cavities filled with bubbly liquids. The model we solve accounts for nonlinearity, dissipation, and dispersion of the two-dimensional media due to the bubbles. The numerical simulations are based on a finite-difference scheme. They consider the bubbles evenly distributed in the liquid. Results are shown for high-amplitude signals. They make it possible to observe how the linear modes turn into multi-frequency nonlinear fields.     

A numerical method for solving the Vlasov–Poisson equation based on the conservative IDO scheme

We have applied the conservative form of the Interpolated Differential Operator (IDO-CF) scheme in order to solve the Vlasov–Poisson equation, which is one of the multi-moment schemes. Through numerical tests of the nonlinear Landau damping and two-stream instability, we compared the present scheme with other schemes such as the Spline and CIP ones. We mainly investigated the conservation property of the L1-norm, energy, entropy and phase space area for each scheme, and demonstrated that the IDO-CF scheme is capable of performing stable long time scale simulation while maintaining high accuracy. The scheme is based on an Eulerian approach, and it can thus be directly used for Fokker–Planck, high dimensional Vlasov–Poisson and also guiding-center drift simulations, aiming at particular problems of plasma physics. The benchmark tests for such simulations have shown that the IDO-CF scheme is superior in keeping the conservation properties without causing serious phase error.     

A Linear Energy-Preserving Finite Volume Element Method for the Improved Korteweg−de Vries Equation

In this paper, we introduce a linearized energy-preserving scheme which preserves the discrete global energy of solutions to the improved Korteweg?deVries equation. The method presented is based on the finite volume element method, by resorting to the variational derivative to transform the improved Korteweg?deVries equation into a new form, and then designing energy-preserving schemes for the transformed equation. The proposed scheme is much more efficient than the standard nonlinear scheme and has good stability. To illustrate its efficiency and conservative properties, we also compare it with other nonlinear schemes. Finally, we verify the efficiency and conservative properties through numerical simulations.     

一类新的差分格式优化目标函数

对差分格式进行优化处理可以提高其谱精度。与高精度(指Taylor展开精度)格式相比,优化格式放大因子的误差随波数的变化不是单调的,而是必然会出现极值点,这样就存在临界距离R_cr,在此距离内优化格式描述的数值波的积累误差小于高精度格式,而超出此距离后优化格式的误差反而大,对于非定常流及气动声学计算来说,控制差分格式的临界距离是必要的。一般的优化目标函数以每个时间推进步的误差为基础(即放大因子法),R_cr不能在优化过程中确定。对此进行分析,指出积累误差的重要性并提出以此为基础的新的优化目标函数,这样在对格式进行优化时可以直接指定临界距离,从而为控制谱精度提供方便。     

Analysis of Nanostructural Waveguides Using Fourth-Order Accurate Finite Difference Methods with Nonuniform Scheme

We successfully apply fourth-order accurate finite difference methods with nonuniform scheme to analysis the symmetric slot waveguides. The results of numerical simulations show that the present nonuniform formula offers the results more accurate than the previously presented second order schemes.     

含时Schrödinger方程的高阶辛FDTD算法研究

提出了一种新的算法——高阶辛时域有限差分法(SFDTD(3, 4): symplectic finite-difference time-domain)求解含时薛定谔方程.在时间上采用三阶辛积分格式离散, 空间上采用四阶精度的同位差分格式离散, 建立了求解含时薛定谔方程的高阶离散辛框架;探讨了高阶辛算法的稳定性及数值色散性.通过理论上的分析及数值算例表明:当空间采用高阶同位差分格式时, 辛积分可提高算法的稳定度;SFDTD(3, 4)法和FDTD(2, 4)法较传统的FDTD(2, 2)法数值色散性明显改善.对二维量子阱和谐振子的仿真结果表明: SFDTD(3, 4)法较传统的FDTD(2, 2)法及高阶FDTD(2, 4)法有着更好的计算精度和收敛性, 且SFDTD(3, 4)法能够保持量子系统的能量守恒, 适用于长时间仿真.     

Finite Difference/Element Method for a Two-Dimensional Modified Fractional Diffusion Equation

We present the finite difference/element method for a two-dimensional modified fractional diffusion equation. The analysis is carried out first for the time semi-discrete scheme, and then for the full discrete scheme. The time discretization is based on the $L1$-approximation for the fractional derivative terms and the second-order backward differentiation formula for the classical first order derivative term. We use finite element method for the spatial approximation in full discrete scheme. We show that both the semi-discrete and full discrete schemes are unconditionally stable and convergent. Moreover, the optimal convergence rate is obtained. Finally, some numerical examples are tested in the case of one and two space dimensions and the numerical results confirm our theoretical analysis.     

Self-trapping and flipping of double-charged vortices in optically induced photonic lattices

We report what is believed to be the first observation of self-trapping and charge-flipping of double-charged optical vortices in two-dimensional photonic lattices. Both on- and off-site excitations lead to the formation of rotating quasi-vortex solitons, reversing the topological charges and the direction of rotation through a quadrupole-like transition state. Experimental results are corroborated with numerical simulations.