A global collisionless PIC code in magnetic coordinates

A global plasma turbulence simulation code, ORB5, is presented. It solves the gyrokinetic electrostatic equations including zonal flows in axisymmetric magnetic geometry. The present version of the code assumes a Boltzmann electron response on magnetic surfaces. It uses a Particle-In-Cell (PIC), δf scheme, 3D cubic B-splines finite elements for the field solver and several numerical noise reduction techniques. A particular feature is the use of straight-field-line magnetic coordinates and a field-aligned Fourier filtering technique that dramatically improves the performance of the code in terms of both the numerical noise reduction and the maximum time step allowed. Another feature is the capability to treat arbitrary axisymmetric ideal MHD equilibrium configurations. The code is heavily parallelized, with scalability demonstrated up to 4096 processors and 10⁹ marker particles. Various numerical convergence tests are performed. The code is validated against an analytical theory of zonal flow residual, geodesic acoustic oscillations and damping, and against other codes for a selection of linear and nonlinear tests.     

Beam-plasma dielectric tensor with Mathematica

We present a Mathematica notebook allowing for the symbolic calculation of the 3×3 dielectric tensor of an electron-beam plasma system in the fluid approximation. Calculation is detailed for a cold relativistic electron beam entering a cold magnetized plasma, and for arbitrarily oriented wave vectors. We show how one can elaborate on this example to account for temperatures, arbitrarily oriented magnetic field or a different kind of plasma.

Program summary

Title of program: TensorCatalog identifier: ADYT_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADYT_v1_0Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandComputer for which the program is designed and others on which it has been tested: Computers: Any computer running Mathematica 4.1. Tested on DELL Dimension 5100 and IBM ThinkPad T42. Installations: ETSI Industriales, Universidad Castilla la Mancha, Ciudad Real, SpainOperating system under which the program has been tested: Windows XP ProProgramming language used: Mathematica 4.1Memory required to execute with typical data: 7.17 MbytesNo. of bytes in distributed program, including test data, etc.: 33 439No. of lines in distributed program, including test data, etc.: 3169Distribution format: tar.gzNature of the physical problem: The dielectric tensor of a relativistic beam plasma system may be quite involved to calculate symbolically when considering a magnetized plasma, kinetic pressure, collisions between species, and so on. The present Mathematica notebook performs the symbolic computation in terms of some usual dimensionless variables.Method of solution: The linearized relativistic fluid equations are directly entered and solved by Mathematica to express the first-order expression of the current. This expression is then introduced into a combination of Faraday and Ampère-Maxwell's equations to give the dielectric tensor. Some additional manipulations are needed to express the result in terms of the dimensionless variables.Restrictions on the complexity of the problem: Temperature effects are limited to small, i.e. non-relativistic, temperatures. The kinetic counterpart of the present Mathematica will usually not compute the required integrals.Typical running time: About 1 minute on a Intel Centrino 1.5 GHz Laptop with 512 MB of RAM.Unusual features of the program: None.     

Auto-Bäcklund transformation and exact solutions of the generalized variable-coefficient Kadomtsev-Petviashvili equation

Based on the idea of the homogeneous balance (HB) method, an auto-Bäcklund transformation (BT) to the generalized variable-coefficient Kadomtsev-Petviashvili (GvcKP) equation is obtained with symbolic computation. By the use of the auto-BT and the ε-expansion method, we can obtain a soliton-like solution including N-solitary wave of the GvcKP equation. Especially, we get a soliton-like solution including two-solitary wave as an illustrative example in detail. Since the cylindrical KP (cKP) equation, the generalized cKP (GcKP) equation and the spherical KP (SKP) equation are all special cases of the GvcKP equation, we can also obtain the corresponding results of these equations respectively.     

A full-wave solver of the Maxwell's equations in 3D cold plasmas

A new solver for Maxwell's equations in three-dimensional (3D) plasma configurations is presented. The new code LEMan (Low-frequency ElectroMagnetic wave propagation) determines a global solution of the wave equation in a realistic stellarator geometry at low frequencies. The code is aimed at the applications with relatively small computational resources and is very efficient in the Alfvén frequency range. In the present work, the cold plasma model is implemented. Finite elements are applied for the radial discretization and the spectral representation is used for the poloidal and toroidal angles. Special care is taken to avoid the numerical pollution of the spectrum as well as to ensure the energy conservation. The numerical scheme and the convergence properties are discussed. Several benchmarks and results in different geometries are presented.     

Conservative global gyrokinetic toroidal full-f five-dimensional Vlasov simulation

A new conservative global gyrokinetic toroidal full-f five-dimensional Vlasov simulation (GT5D) is developed using a novel non-dissipative conservative finite difference scheme. The scheme guarantees numerical stability by satisfying relevant first principles in the modern gyrokinetic theory, and enables robust and accurate simulations of tokamak micro-turbulence. GT5D is verified through comparisons of zonal flow damping tests, linear analyses of ion temperature gradient driven (ITG) modes, and nonlinear ITG turbulence simulations against a global gyrokinetic toroidal δf particle code. In the comparison, global solutions of the ITG turbulence are identified quantitatively by using two gyrokinetic codes based on particle and mesh approaches.     

A B-spline Galerkin method for the Dirac equation

The B-spline Galerkin method is first investigated for the simple eigenvalue problem, y^″=−λ²y, that can also be written as a pair of first-order equations y^′=λz, z^′=−λy. Expanding both y(r) and z(r) in the Bk basis results in many spurious solutions such as those observed for the Dirac equation. However, when y(r) is expanded in the Bk basis and z(r) in the dBk/dr basis, solutions of the well-behaved second-order differential equation are obtained. From this analysis, we propose a stable method (Bk,Bk±1) basis for the Dirac equation and evaluate its accuracy by comparing the computed and exact R-matrix for a wide range of nuclear charges Z and angular quantum numbers κ. When splines of the same order are used, many spurious solutions are found whereas none are found for splines of different order. Excellent agreement is obtained for the R-matrix and energies for bound states for low values of Z. For high Z, accuracy requires the use of a grid with many points near the nucleus. We demonstrate the accuracy of the bound-state wavefunctions by comparing integrals arising in hyperfine interaction matrix elements with exact analytic expressions. We also show that the Thomas-Reiche-Kuhn sum rule is not a good measure of the quality of the solutions obtained by the B-spline Galerkin method whereas the R-matrix is very sensitive to the appearance of pseudo-states.     

RacoonWW1.3: A Monte Carlo program for four-fermion production at ee colliders

We present the Monte Carlo generator RacoonWW that computes cross sections to all processes e⁺e⁻→4f and e⁺e⁻→4fγ and calculates the complete electroweak radiative corrections to e⁺e⁻→WW→4f in the electroweak Standard Model in double-pole approximation. The calculation of the tree-level processes e⁺e⁻→4f and e⁺e⁻→4fγ is based on the full matrix elements for massless (polarized) fermions. When calculating radiative corrections to e⁺e⁻→WW→4f, the complete virtual doubly-resonant electroweak corrections are included, i.e. the factorizable and non-factorizable virtual corrections in double-pole approximation, and the real corrections are based on the full matrix elements for e⁺e⁻→4fγ. The matching of soft and collinear singularities between virtual and real corrections is done alternatively in two different ways, namely by using a subtraction method or by applying phase-space slicing. Higher-order initial-state photon radiation and naive QCD corrections are taken into account. RacoonWW also provides anomalous triple gauge-boson couplings for all processes e⁺e⁻→4f and anomalous quartic gauge-boson couplings for all processes e⁺e⁻→4fγ.     

Tension spline approach for the numerical solution of nonlinear Klein-Gordon equation

The nonlinear Klein-Gordon equation describes a variety of physical phenomena such as dislocations, ferroelectric and ferromagnetic domain walls, DNA dynamics, and Josephson junctions. We derive approximate expressions for the dispersion relation of the nonlinear Klein-Gordon equation in the case of strong nonlinearities using a method based on the tension spline function and finite difference approximations. The resulting spline difference schemes are analyzed for local truncation error, stability and convergence. It has been shown that by suitably choosing the parameters, we can obtain two schemes of O(k²+k²h²+h²) and O(k²+k²h²+h⁴). In the end, some numerical examples are provided to demonstrate the effectiveness of the proposed schemes.     

Implementation of the CIP algorithm to magnetohydrodynamic simulations

An implementation of the Constrained Interpolation Profile (CIP) algorithm to magnetohydrodynamic (MHD) simulations is presented. First we transform the original momentum and magnetic induction equations to unfamiliar forms by introducing Elsässer variables [W.M. Elsässer, The hydromagnetic equations, Phys. Rev. (1950)]. In this formulation, while the compressional and pressure gradient terms remain as non-advective terms, the advective and magnetic stress terms are expressed in the form of an advection equation, which enables us to use the CIP algorithm. We have examined some 1D test problems using the code based on this formula. Linear Alfvén wave propagation tests reveal that the developed code is capable of solving any Alfvén wave propagation with only small numerical diffusion and phase errors up to k?h=2.5 (where ?h is the grid spacing). A shock tube test shows good agreement with a previous result with less numerical oscillation at the shock front and the contact discontinuity which are captured within a few grid points. Extension of the 1D code to the multi-dimensional case is straightforward. We have calculated the 3D nonlinear evolution of the Kelvin-Helmholtz instability (KHI) and compared the result with our previous study. We find that our new MHD code is capable of following the 3D turbulence excited by the KHI while retaining the solenoidal property of the magnetic field.     

Fitting sparse multidimensional data with low-dimensional terms

An algorithm that fits a continuous function to sparse multidimensional data is presented. The algorithm uses a representation in terms of lower-dimensional component functions of coordinates defined in an automated way and also permits dimensionality reduction. Neural networks are used to construct the component functions.

Program summary

Program title: RS_HDMR_NNCatalogue identifier: AEEI_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEEI_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 19 566No. of bytes in distributed program, including test data, etc.: 327 856Distribution format: tar.gzProgramming language: MatLab R2007bComputer: any computer running MatLabOperating system: Windows XP, Windows Vista, UNIX, LinuxClassification: 4.9External routines: Neural Network Toolbox Version 5.1 (R2007b)Nature of problem: Fitting a smooth, easily integratable and differentiatable, function to a very sparse (∼2-3 points per dimension) multidimensional (D?6) large (∼¹⁰⁴-¹⁰⁵ data) dataset.Solution method: A multivariate function is represented as a sum of a small number of terms each of which is a low-dimensional function of optimised coordinates. The optimal coordinates reduce both the dimensionality and the number of the terms. Neural networks (including exponential neurons) are used to obtain a general and robust method and a functional form which is easily differentiated and integrated (in the case of exponential neurons).Running time: Depends strongly on the dataset to be modelled and the chosen structure of the approximating function, ranges from about a minute for ∼¹⁰³ data in 3-D to about a day for ∼¹⁰⁵ data in 15-D.     

A global optimization method for Landau gauge fixing in lattice QCD

An algorithm for gauge fixing to the Landau gauge in the fundamental modular region in lattice QCD is described. The method, a combination of an evolutionary algorithm with a steepest descent method, is able to solve the problem of the nonperturbative gauge fixing. The performance of the combined algorithm is investigated on 8⁴, β=5.7, and 16⁴, β=6.0, lattice SU(3) gauge configurations.     

Applications of complete discrimination system for polynomial for classifications of traveling wave solutions to nonlinear differential equations

If a partial differential equation is reduced to an ordinary differential equation in the form u^′(ξ)=G(u,θ₁,…,θm) under the traveling wave transformation, where θ₁,…,θm are parameters, its solutions can be written as an integral form . Therefore, the key steps are to determine the parameters' scopes and to solve the corresponding integral. When G is related to a polynomial, a mathematical tool named complete discrimination system for polynomial is applied to this problem so that the parameter's scopes can be determined easily. The complete discrimination system for polynomial is a natural generalization of the discrimination △=b²−4ac of the second degree polynomial ax²+bx+c. For example, the complete discrimination system for the third degree polynomial F(w)=w³+d₂w²+d₁w+d₀ is given by and . In the paper, we give some new applications of the complete discrimination system for polynomial, that is, we give the classifications of traveling wave solutions to some nonlinear differential equations through solving the corresponding integrals. In finally, as a result, we give a partial answer to a problem on Fan's expansion method.     

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.     

QQ-onia package: a numerical solution to the Schrödinger radial equation for heavy quarkonium

This paper presents the basics of the QQ-onia package, a software based upon the Numerov O(h⁶) method which can be used to solve the Schrödinger radial equation using a suitable potential V(r) for the heavy quarkonium system. This package also allows the analysis of relevant properties of those resonances such as the square of the wave functions at the origin, their corresponding derivatives for l≠0 states, typical heavy-quark velocities, and mean square radii. Besides, it includes a tool to analyze the spin dependent contributions to the heavy quarkonia spectrum, providing the splitting of n³S₁−n¹S₀, as well as the n³PJ−n¹P₁ energy levels. Finally a simple software implemented in QQ-onia to compute E1 transition rates is presented.

Program summary

Program title: QQ-onia packageCatalogue identifier: AECQ_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AECQ_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 17 706No. of bytes in distributed program, including test data, etc.: 2 334 506Distribution format: tar.gzProgramming language: PAW, Physics Analysis Workstation (http://wwwasd.web.cern.ch/wwwasd/paw/)Computer: PC/WorkstationOperating system: Windows-XX and Unix (Linux)Classification: 11.1, 11.6Nature of problem: Software to solve the Schrödinger radial equation using a suitable potential V(r) for the heavy quarkonium system, allowing to perform spectroscopy. It also allows the analysis of relevant quantities of those resonances such as the square of the wave functions at the origin, their corresponding derivatives for l≠0 states, typical heavy-quark velocities, and mean square radii. The package is a (userfriendly) multipurpose tool for dealing with different heavy quarkonium systems, providing a way to study the influence of a given potential on a series of relevant physical quantities, by either varying parameterized values of a well-known potential form, or by including new terms.Solution method: Based upon the Numerov O(h⁶) method, we perform a matching procedure to the reduced wave function at the cut point. We also perform a normalization technique for these wave functions taking into account the different domains when we use a Numerov backward-forward technique. In the case of l?2 we present a way to find the corresponding derivatives at the origin by only calculating the reduced radial wave function and first derivative. When estimating the heavy quark velocity, we introduce an additional way to compute this quantity from the virial theorem. The calculated reduced wave functions and radial wave functions at the origin are later used to obtain the heavy quarkonia nL splitting and E1 transition rates.Additional comments: Using Windows, to optimize the edition of the files, please, open it with MFC-WORDPAD.Running time: It depends on the choice of the r range, and the number of energy steps.     

Morphismes et bimorphismes d'arbres

For every pair of positive integers n and p, there is a language accepted by a real-time deterministic pushdown automaton with n states and p stack symbols and size O(np), for which every context-free grammar needs at least n²p+1 nonterminals if n>1 (or p non-terminals if n = 1). It follows that there are context-free languages which can be recognized by pushdown automata of size O(np), but which cannot be generated by context-free grammars of size smaller than O(n²p); and that the standard construction for converting a pushdown automaton to a context-free grammar is optimal in the sense that it infinitely often produces grammars with the fewest number of nonterminals possible.     

Studies of angular correlations in the decays Bs→J/ψφ by using the SIMUB generator

The performance of the method of angular moments on the ΔΓ_s determination from analysis of untagged decays is examined by using the SIMUB generator. The results of Monte Carlo studies with evaluation of measurement errors are presented. The method of angular moments gives stable results for the estimate of ΔΓ_s and is found to be an efficient and flexible tool for the quantitative investigation of the B⁰_s→J/ψφ decay. The statistical error of the ratio ΔΓ_s/Γ_s for values of this ratio in the interval [0.03,0.3] was found to be independent on this value, being 0.015 for 10⁵ events.     

Exact analytical expressions and numerical analysis of two-center Franck-Condon factors and matrix elements over displaced harmonic oscillator wave functions

A detailed study is undertaken, using various techniques, in deriving analytical formula of Franck-Condon overlap integrals and matrix elements of various functions of power (xl), exponential (exp(−2cx)) and Gaussian (exp(−cx²)) over displaced harmonic oscillator wave functions with arbitrary frequencies. The results suggested by previous experience with various algorithms are presented in mathematically compact form and consist of generalization. The relationships obtained are valid for the arbitrary values of parameters and the computation results are in good agreement with the literature. The numerical results illustrate clearly a further reduction in calculation times.

Program summary

Program name:FRANCKCatalogue identifier:ADXX_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADXX_v1_0Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandProgramming language:Mathematica 5.0Computer:Pentium M 1.4 GHzOperating system:Mathematica 5.0RAM:512 MBNo. of lines in distributed program, including test data, etc.:825No. of bytes in distributed program, including test data, etc.:16 344Distribution format:tar.gzNature of problem:The programs calculate the Franck-Condon factors and matrix elements over displaced harmonic oscillator wave functions with arbitrary quantum numbers (n,n₁), frequencies (a,a₁) and displacement (d) for the various functions of power (xl), exponential (exp(−2cx)) and Gaussian (exp(−cx²)).Solution method:The Franck-Condon factors and matrix elements are evaluated using binomial coefficients and basic integrals.Restrictions:The results obtained by the present programs show great numerical stability for arbitrary quantum numbers (n,n₁), frequencies (a,a₁) and displacement (d).Unusual features:NoneRunning time:As an example, for the value of Franck-Condon Overlap Integral Inn^′(d;α,α^′)=0.004405001887372332 with n=3, n₁=2, a=4, a₁=3, d=2, the compilation time in a Pentium M 1.4 GHz computer is 0.18 s. Execution time depends on the values of integral parameters n, n^′, d, α, α^′.     

L1PMA: A Fortran 77 package for best L1 piecewise monotonic data smoothing

Fortran 77 software is presented for the calculation of a best L₁ approximation to n measurements that include random errors by requiring k−1 sign changes in the first divided differences of the approximation or equivalently k monotonic sections, alternately increasing and decreasing. A dynamic programming algorithm separates the measurements into optimal disjoint sections of adjacent data and applies to each section a single L₁ monotonic calculation. The most distinctive feature of the algorithm is that it terminates at a global minimum in at most n³+O(kn²) computer operations, although this calculation can exhibit O(n^k) local minima, because the optimal positions of the turning points are also unknowns of the optimization process. The arithmetic operations involved in this calculation are comparisons mainly spent in finding the medians of subranges of data during the monotonic calculations. The package employs techniques for median and for best L₁ monotonic approximation, while full details of these techniques are specified. The package has been applied and tested on a variety of data that have substantial differences and showed quadratic behaviour in n. Some numerical results demonstrate the performance of the method. Further, there is a commentary on the division of the code into subroutines. Driver programs and numerical examples with output are provided to help new users of the method. Besides that piecewise monotonicity is a property of a wide range of functions, an important application of the method is in estimating turning points of a function from some noisy measurements of its values.     

Development of a Geant4 solid for stereo mini-jet cells in a cylindrical drift chamber

Stereo mini-jet cells will be indispensable components of a future e⁺e⁻ linear collider central tracker such as JLC-CDC. There is, however, no official Geant4 solid available at present to describe such geometrical objects, which had been a major obstacle for us to develop a full Geant4-based simulator with stereo cells built in. We have thus extended Geant4 to include a new solid (TwistedTubs), which consists of three kinds of surfaces: two end planes, inner and outer hyperboloidal surfaces, and two so-called twisted surfaces that make slant and twisted φ-boundaries. Design philosophy and its realization in the Geant4 framework are described together with algorithmic details. We have implemented stereo cells with the new solid, and tested them using geantinos and Pythia events (e⁺e⁻→ZH at  GeV). The performance was found reasonable: the stereo cells consumed only 25% more CPU time than ordinary axial cells.