Comparison of time splitting and backsubstitution methods for integrating Vlasov's equation with magnetic fields

The standard approach for integrating the multidimensional Vlasov equation using grid based, conservative schemes is based on a time splitting approach. Here, we show that although the truncation error is of second order, time splitting can introduce systematic heating of the plasma. We introduce a backsubstitution method, which not only avoids this deficiency but also is computationally less expensive. The general approach is demonstrated in conjunction with Boris' scheme for evaluating the characteristics.     

Comparison of Eulerian Vlasov solvers

Vlasov methods, which instead of following the particle trajectories, solve directly the Vlasov equation on a grid of phase space have proven to be an efficient alternative to the Particle-In-Cell method for some specific problems. Such methods are useful, in particular, to obtain high precision in regions where the distribution function is small.Gridded Vlasov methods have the advantage of being completely free of numerical noise, however the discrete formulations contain some other numerical artifacts, like diffusion or dissipation. We shall compare in this paper different types of methods for solving the Vlasov equation on a grid in phase space: the semi-Lagrangian method, the finite volume method, the spectral method, and a method based on a finite difference scheme, conserving exactly several invariants of the system. Moreover, for each of those classes of methods, we shall first compare different interpolation or reconstruction procedures. Then we shall investigate the cost in memory as well as in CPU time which is a very important issue because of the size of the problem defined on the phase space.     

The parallel implementation of the one-dimensional Fourier transformed Vlasov-Poisson system

A parallel implementation of an algorithm for solving the one-dimensional, Fourier transformed Vlasov-Poisson system of equations is documented, together with the code structure, file formats and settings to run the code. The properties of the Fourier transformed Vlasov-Poisson system is discussed in connection with the numerical solution of the system. The Fourier method in velocity space is used to treat numerical problems arising due the filamentation of the solution in velocity space. Outflow boundary conditions in the Fourier transformed velocity space removes the highest oscillations in velocity space. A fourth-order compact Padé scheme is used to calculate derivatives in the Fourier transformed velocity space, and spatial derivatives are calculated with a pseudo-spectral method. The parallel algorithms used are described in more detail, in particular the parallel solver of the tri-diagonal systems occurring in the Padé scheme.

Program summary

Title of program:vlasovCatalogue identifier:ADVQProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADVQProgram obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandOperating system under which the program has been tested: Sun Solaris; HP-UX; Read Hat LinuxProgramming language used: FORTRAN 90 with Message Passing Interface (MPI)Computers: Sun Ultra Sparc; HP 9000/785; HP IPF (Itanium Processor Family) ia64 Cluster; PCs clusterNumber of lines in distributed program, including test data, etc.:3737Number of bytes in distributed program, including test data, etc.:18 772Distribution format: tar.gzNature of physical problem: Kinetic simulations of collisionless electron-ion plasmas.Method of solution: A Fourier method in velocity space, a pseudo-spectral method in space and a fourth-order Runge-Kutta scheme in time.Memory required to execute with typical data: Uses typically of the order 10⁵-10⁶ double precision numbers.Restriction on the complexity of the problem: The program uses periodic boundary conditions in space.Typical running time: Depends strongly on the problem size, typically few hours if only electron dynamics is considered and longer if both ion and electron dynamics is important.Unusual features of the program: No     

A semi-implicit Hall-MHD solver using whistler wave preconditioning

The dispersive character of the Hall-MHD solutions, in particular the whistler waves, is a strong restriction to numerical treatments of this system. Numerical stability demands a time step dependence of the form Δt∝²(Δx) for explicit calculations. A new semi-implicit scheme for integrating the induction equation is proposed and applied to a reconnection problem. It is based on a fix point iteration with a physically motivated preconditioning. Due to its convergence properties, short wavelengths converge faster than long ones, thus it can be used as a smoother in a nonlinear multigrid method.     

A semi-Lagrangian code for nonlinear global simulations of electrostatic drift-kinetic ITG modes

A semi-Lagrangian code for the solution of the electrostatic drift-kinetic equations in straight cylinder configuration is presented. The code, CYGNE, is part of a project with the long term aim of studying microturbulence in fusion devices. The code has been constructed in such a way as to preserve a good control of the constants of motion, possessed by the drift-kinetic equations, until the nonlinear saturation of the ion-temperature-gradient modes occurs. Studies of convergence with phase space resolution and time-step are presented and discussed. The code is benchmarked against electrostatic Particle-in-Cell codes.     

Mathematical modeling of blood circulation system and its practical application

The human blood circulation system is represented by a nonlinear oscillation system for computer-aided digital modeling in real time scale. A parametric identification problem is formulated and its numerical solution algorithm is designed. A computer-aided blood circulation modeling and identification system is designed. The new approaches to construct real control systems for artificial and auxiliary blood circulation elements are based on neurocomputer technologies.     

The nonlinear gyro-kinetic flux tube code GKW

A new nonlinear gyro-kinetic flux tube code (GKW) for the simulation of micro instabilities and turbulence in magnetic confinement plasmas is presented in this paper. The code incorporates all physics effects that can be expected from a state of the art gyro-kinetic simulation code in the local limit: kinetic electrons, electromagnetic effects, collisions, full general geometry with a coupling to a MHD equilibrium code, and E×B shearing. In addition the physics of plasma rotation has been implemented through a formulation of the gyro-kinetic equation in the co-moving system. The gyro-kinetic model is five-dimensional and requires a massive parallel approach. GKW has been parallelised using MPI and scales well up to 8192+ cores. The paper presents the set of equations solved, the numerical methods, the code structure, and the essential benchmarks.

Program summary

Program title: GKWCatalogue identifier: AEES_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEES_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: GNU GPL v3No. of lines in distributed program, including test data, etc.: 29 998No. of bytes in distributed program, including test data, etc.: 206 943Distribution format: tar.gzProgramming language: Fortran 95Computer: Not computer specificOperating system: Any for which a Fortran 95 compiler is availableHas the code been vectorised or parallelised?: Yes. The program can efficiently utilise 8192+ processors, depending on problem and available computer. 128 processors is reasonable for a typical nonlinear kinetic run on the latest x86-64 machines.RAM:∼128 MB–1 GB for a linear run; 25 GB for typical nonlinear kinetic run (30 million grid points)Classification: 19.8, 19.9, 19.11External routines: None required, although the functionality of the program is somewhat limited without a MPI implementation (preferably MPI-2) and the FFTW3 library.Nature of problem: Five-dimensional gyro-kinetic Vlasov equation in general flux tube tokamak geometry with kinetic electrons, electro-magnetic effects and collisionsSolution method: Pseudo-spectral and finite difference with explicit time integrationAdditional comments: The MHD equilibrium code CHEASE [1] is used for the general geometry calculations. This code has been developed in CRPP Lausanne and is not distributed together with GKW, but can be downloaded separately. The geometry module of GKW is based on the version 7.1 of CHEASE, which includes the output for Hamada coordinates.Running time: (On recent x86-64 hardware) ∼10 minutes for a short linear problem; 48 hours for typical nonlinear kinetic run.References:

•  

  [1] H. Lütjens, A. Bondeson, O. Sauter, Comput. Phys. Comm. 97 (1996) 219, http://cpc.cs.qub.ac.uk/summaries/ADDH_v1_0.html.

     

A transport simulation code for inertial confinement fusion relevant laser-plasma interaction

We present a code for the simulation of laser-plasma interaction processes relevant for applications in inertial confinement fusion. The code consists of a fully nonlinear hydrodynamics in two spatial dimensions using a Lagrangian, discontinuous Galerkin-type approach, a paraxial treatment of the laser field and a spectral treatment of the dominant non-local transport terms. The code is fully parallelized using MPI in order to be able to simulate macroscopic plasmas.One example of a fully nonlinear evolution of a laser beam in an underdense plasma is presented for the conditions previewed for the future MegaJoule laser project.     

Theory of dynamic entropy-operator systems and its applications

The phenomenological and mathematical definitions of a class of dynamic systems with an entropy operator are formulated. Dynamic systems with an entropy operator are classified and the main theoretical results pertaining to the properties of entropy operators and these dynamic systems are studied within this classification. By way of examples, restoration of monochromatic images and modeling of labor market are examined.     

A new simulation code for particle diffusion in anisotropic, large-scale and turbulent magnetic fields

A new Monte Carlo code is presented that simulates the scattering processes of energetic particles in turbulent magnetic fields. The growing number of analytical models for anisotropic turbulence geometries gives rise to the need of fast and adaptable simulation codes that, in order to be able to judge the accuracy of the results, calculate the estimated mean errors of the transport parameters. Furthermore, the need to understand the interplay of scattering in anisotropic large-scale (such as the solar magnetic field) and turbulent (such as the Maltese cross-like structured solar wind turbulence) magnetic fields is accounted for with the calculation of the off-diagonal elements of the diffusion tensor.     

Extended Jacobian elliptic function algorithm with symbolic computation to construct new doubly-periodic solutions of nonlinear differential equations

With the aid of computerized symbolic computation, the extended Jacobian elliptic function expansion method and its algorithm are presented by using some relations among ten Jacobian elliptic functions and are very powerful to construct more new exact doubly-periodic solutions of nonlinear differential equations in mathematical physics. The new (2+1)-dimensional complex nonlinear evolution equations is chosen to illustrate our algorithm such that sixteen families of new doubly-periodic solutions are obtained. When the modulus m→1 or 0, these doubly-periodic solutions degenerate as solitonic solutions including bright solitons, dark solitons, new solitons as well as trigonometric function solutions.     

On the evaluation of Fermi-Dirac integral and its derivatives by IMT and DE quadrature methods

The numerical values of the derivatives of the Fermi-Dirac integral, up to the third order, with respect to its parameters are evaluated by both the IMT and the Double Exponential integration methods. These integrals find extensive use in astrophysical problems. It is found that both the IMT and the DE methods are very efficient, especially for handling the end point integrable singularities. Also, the DE method is slightly superior to the IMT method. Unlike the IMT or Gauss methods, the nodes of DE method can be evaluated in terms of built in elementary functions, which is a big advantage.     

New travelling wave solutions to modified CH and DP equations

A new procedure for finding exact travelling wave solutions to the modified Camassa-Holm and Degasperis-Procesi equations is proposed. It turns out that many new solutions are obtained. Furthermore, these solutions are in general forms, and many known solutions to these two equations are only special cases of them.     

A basis-set based Fortran program to solve the Gross-Pitaevskii equation for dilute Bose gases in harmonic and anharmonic traps

Inhomogeneous boson systems, such as the dilute gases of integral spin atoms in low-temperature magnetic traps, are believed to be well described by the Gross-Pitaevskii equation (GPE). GPE is a nonlinear Schrödinger equation which describes the order parameter of such systems at the mean field level. In the present work, we describe a Fortran 90 computer program developed by us, which solves the GPE using a basis set expansion technique. In this technique, the condensate wave function (order parameter) is expanded in terms of the solutions of the simple-harmonic oscillator (SHO) characterizing the atomic trap. Additionally, the same approach is also used to solve the problems in which the trap is weakly anharmonic, and the anharmonic potential can be expressed as a polynomial in the position operators x, y, and z. The resulting eigenvalue problem is solved iteratively using either the self-consistent-field (SCF) approach, or the imaginary time steepest-descent (SD) approach. Iterations can be initiated using either the simple-harmonic-oscillator ground state solution, or the Thomas-Fermi (TF) solution. It is found that for condensates containing up to a few hundred atoms, both approaches lead to rapid convergence. However, in the strong interaction limit of condensates containing thousands of atoms, it is the SD approach coupled with the TF starting orbitals, which leads to quick convergence. Our results for harmonic traps are also compared with those published by other authors using different numerical approaches, and excellent agreement is obtained. GPE is also solved for a few anharmonic potentials, and the influence of anharmonicity on the condensate is discussed. Additionally, the notion of Shannon entropy for the condensate wave function is defined and studied as a function of the number of particles in the trap. It is demonstrated numerically that the entropy increases with the particle number in a monotonic way.

Program summary

Title of program:bose.xCatalogue identifier:ADWZ_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADWZ_v1_0Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format:tar.gzComputers:PC's/Linux, Sun Ultra 10/Solaris, HP Alpha/Tru64, IBM/AIXProgramming language used:mostly Fortran 90Number of bytes in distributed program, including test data, etc.:27 313Number of lines in distributed program, including test data, etc.:28 015Card punching code:ASCIINature of physical problem:It is widely believed that the static properties of dilute Bose condensates, as obtained in atomic traps, can be described to a fairly good accuracy by the time-independent Gross-Pitaevskii equation. This program presents an efficient approach of solving this equation.Method of solution:The solutions of the Gross-Pitaevskii equation corresponding to the condensates in atomic traps are expanded as linear combinations of simple-harmonic oscillator eigenfunctions. Thus, the Gross-Pitaevskii equation which is a second-order nonlinear differential equation, is transformed into a matrix eigenvalue problem. Thereby, its solutions are obtained in a self-consistent manner, using methods of computational linear algebra.Unusual features of the program:None     

The Monte Carlo Event Generator AcerMC version 1.0 with interfaces to PYTHIA 6.2 and HERWIG 6.3

The AcerMC Monte Carlo Event Generator is dedicated for the generation of Standard Model background processes at pp LHC collisions. The program itself provides a library of the massive matrix elements and phase space modules for generation of a set of selected processes: , , , , and complete electroweak process. The hard process event, generated with one of these modules, can be completed by the initial and final state radiation, hadronization and decays, simulated with either PYTHIA or HERWIG Monte Carlo event generator. Interfaces to both of these generators are provided in the distribution version. The matrix element codes have been derived with the help of the MADGRAPH package. The phase-space generation is based on the multi-channel self-optimizing approach as proposed in NEXTCALIBUR event generator. Eventually, additional smoothing of the phase space was obtained by using a modified ac-VEGAS routine in order to improve the generation efficiency.     

Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics

We discuss several applications of the recently proposed combined nonlinear-condensation transformation (CNCT) for the evaluation of slowly convergent, nonalternating series. These include certain statistical distributions which are of importance in linguistics, statistical-mechanics theory, and biophysics (statistical analysis of DNA sequences). We also discuss applications of the transformation in experimental mathematics, and we briefly expand on further applications in theoretical physics. Finally, we discuss a related Mathematica program for the computation of Lerch's transcendent.     

GPRS网络下视频信息的码率优化策略

鉴于GPRS网络下行链路的平均速度仅25Kbps左右,我们设计并实施了一个切实、有效的视频信息码率优化策略.据此建立的应用系统从不同类型原始流媒体素材的固有特征出发,自适应地选择视频编码参数,使最终形成的音视频码率适合GPRS网络带宽,以获得良好的收视效果.     

Filter diagonalization: Filtering and postprocessing with prolates

A detailed account is given of a recent modification of the Filter Diagonalization technique that serves to analyze a signal spectrum within a selected energy range. Our approach employs for filtering the eigenfunctions of the Finite Fourier Transform, or prolates, which are superior to other filters due to their special properties. In particular, prolates are simultaneously band-limited and highly concentrated at a finite time-interval, producing filters with optimal accuracy. In addition both features are acquired by the convolution of a band-limited function with a prolate, that permits the latter to be interpolated via the Walter and Shen sampling formula, which essentially simplifies the supplementary computations. Rigorous filtering error estimates are obtained. Test calculations illustrate the facilities of the presented modification.     

Finite difference approach for the two-dimensional Schrödinger equation with application to scission-neutron emission

We present a grid-based procedure to solve the eigenvalue problem for the two-dimensional Schrödinger equation in cylindrical coordinates. The Hamiltonian is discretized by using adapted finite difference approximations of the derivatives and this leads to an algebraic eigenvalue problem with a large (sparse) matrix, which is solved by the method of Arnoldi. By this procedure the single particle eigenstates of nuclear systems with arbitrary deformations can be obtained. As an application we have considered the emission of scission neutrons from fissioning nuclei.     

A new approach to code motion and its application to hoisting

An approach to code motion and hoisting, a program optimization technique, is discussed. The safety and profitability of optimization in general and hoisting in particular are analyzed. By restricting the analysis to a spanning tree imposed on the program graph, a linear algorithm is developed that provides sufficient but not necessary conditions for hoisting.