Implicit methods for fluid dynamics

This paper represents the contributions to the development of implicit procedures for solving the equations of fluid dynamics made by Briley and McDonald (1975)[1], Beam and Warming (1976, 1978) [2] and [3] and Lombard et al. (1983) [4]. The contributions of Briley and McDonald and Beam and Warming are well known, but Lombard has not been fully recognized for his innovative contributions to flux vector splitting and use of the DDADI (Diagonally Dominate Alternating Direction Implicit) algorithm. Their contributions are presented herein.Fully implicit algorithms are applied to two complex flow problems of current interest, (1) hypersonic non-equilibrium flow about a blunt nosed body and (2) flow within an MFD (magneto-fluid dynamics) accelerator. These two applications would be exceeding costly without the use of fully implicit methods.     

Stability analysis in spanwise-periodic double-sided lid-driven cavity flows with complex cross-sectional profiles

Three-dimensional linear instability analyses are presented of steady two-dimensional laminar flows in the lid-driven cavity defined by [15] and further analyzed in the present volume [1], as well as in a derivative of the same geometry. It is shown that in both of the geometries considered three-dimensional BiGlobal instability leads to deviation of the flow from the two-dimensional solution; the analysis results are used to define low- and high-Reynolds number solutions by reference to the flow physics. Critical conditions for linear global instability and neutral loops are presented in both geometries.     

S-states of helium-like ions

A simple Mathematica (version 7) code for computing S-state energies and wave functions of two-electron (helium-like) ions is presented. The elegant technique derived from the classical papers of Pekeris (1958, 1959, 1962, 1965, 1971) [1], [2] and [3] is applied. The basis functions are composed of the Laguerre functions. The method is based on the perimetric coordinates and specific properties of the Laguerre polynomials. Direct solution of the generalized eigenvalues and eigenvectors problem is used, distinct from the Pekeris works. No special subroutines were used, only built-in objects supported by Mathematica. The accuracy of the results and computation times depend on the basis size. The ground state and the lowest triplet state energies can be computed with a precision of 12 and 14 significant figures, respectively. The accuracy of the higher excited states calculations is slightly worse. The resultant wave functions have a simple analytical form, that enables calculation of expectation values for arbitrary physical operators without any difficulties. Only three natural parameters are required in the input.The above Mathematica code is simpler than the earlier version (Liverts and Barnea, 2010 [4]). At the same time, it is faster and more accurate.

Program summary

Program title: TwoElAtomSL(SH)Catalogue identifier: AEHY_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEHY_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.: 11 434No. of bytes in distributed program, including test data, etc.: 540 063Distribution format: tar.gzProgramming language: Mathematica 7.0Computer: Any PCOperating system: Any which supports Mathematica; tested under Microsoft Windows XP and Linux SUSE 11.0RAM:?¹⁰⁹ bytesClassification: 2.1, 2.2, 2.7, 2.9Nature of problem: The Schrödinger equation for atoms (ions) with more than one electron has not been solved analytically. Approximate methods must be applied in order to obtain the wave functions or another physical attributes from quantum mechanical calculations.Solution method: The S-wave function is expanded into a triple set of basis functions which are composed of the exponentials combined with the Laguerre polynomials in the perimetric coordinates. Using specific properties of the Laguerre polynomials, solution of the two-electron Schrödinger equation reduces to solving the generalized eigenvalues and eigenvector problem for the proper Hamiltonian. The unknown exponential parameter is determined by means of minimization of the corresponding eigenvalue (energy).Restrictions: First, the too large length of expansion (basis size) takes the too large computation time and operative memory giving no perceptible improvement in accuracy. Second, the number of shells Ω in the wave function expansion enables one to calculate the excited nS-states up to n=Ω+1 inclusive.Running time: 2–60 minutes (depends on basis size and computer speed).     

Improvements of the particle-in-cell code EUTERPE for petascaling machines

In the present work we report some performance measures and computational improvements recently carried out using the gyrokinetic code EUTERPE (Jost, 2000 [1] and Jost et al., 1999 [2]), which is based on the general particle-in-cell (PIC) method. The scalability of the code has been studied for up to sixty thousand processing elements and some steps towards a complete hybridization of the code were made. As a numerical example, non-linear simulations of Ion Temperature Gradient (ITG) instabilities have been carried out in screw-pinch geometry and the results are compared with earlier works. A parametric study of the influence of variables (step size of the time integrator, number of markers, grid size) on the quality of the simulation is presented.     

Confronting intractability via parameters

One approach to confronting computational hardness is to try to understand the contribution of various parameters to the running time of algorithms and the complexity of computational tasks. Almost no computational tasks in real life are specified by their size alone. It is not hard to imagine that some parameters contribute more intractability than others and it seems reasonable to develop a theory of computational complexity which seeks to exploit this fact. Such a theory should be able to address the needs of practitioners in algorithmics. The last twenty years have seen the development of such a theory. This theory has a large number of successes in terms of a rich collection of algorithmic techniques, both practical and theoretical, and a fine-grained intractability theory. Whilst the theory has been widely used in a number of areas of applications including computational biology, linguistics, VLSI design, learning theory and many others, knowledge of the area is highly varied. We hope that this article will show the basic theory and point at the wide array of techniques available. Naturally the treatment is condensed, and the reader who wants more should go to the texts of Downey and Fellows (1999) [2], Flum and Grohe (2006) [59], Niedermeier (2006) [28], and the upcoming undergraduate text (Downey and Fellows 2012) [278].     

A new approach to Lipschitz continuity in state constrained optimal control

For a linear-quadratic state constrained optimal control problem, it is proved in [11] that under an independence condition for the active constraints, the optimal control is Lipschitz continuous. We now give a new proof of this result based on an analysis of the Euler discretization given in [9]. There we exploit the Lipschitz continuity of the control to estimate the error in the Euler discretization. Here we show that the theory developed for the Euler discretization can be used to derive the Lipschitz continuity of the optimal control.     

Generalised theory on asymptotic stability and boundedness of stochastic functional differential equations

Asymptotic stability and boundedness have been two of most popular topics in the study of stochastic functional differential equations (SFDEs) (see e.g. Appleby and Reynolds (2008), Appleby and Rodkina (2009), Basin and Rodkina (2008), Khasminskii (1980), Mao (1995), Mao (1997), Mao (2007), Rodkina and Basin (2007), Shu, Lam, and Xu (2009), Yang, Gao, Lam, and Shi (2009), Yuan and Lygeros (2005) and Yuan and Lygeros (2006)). In general, the existing results on asymptotic stability and boundedness of SFDEs require (i) the coefficients of the SFDEs obey the local Lipschitz condition and the linear growth condition; (ii) the diffusion operator of the SFDEs acting on a C^2,1-function be bounded by a polynomial with the same order as the C^2,1-function. However, there are many SFDEs which do not obey the linear growth condition. Moreover, for such highly nonlinear SFDEs, the diffusion operator acting on a C^2,1-function is generally bounded by a polynomial with a higher order than the C^2,1-function. Hence the existing criteria on stability and boundedness for SFDEs are not applicable and we see the necessity to develop new criteria. Our main aim in this paper is to establish new criteria where the linear growth condition is no longer needed while the up-bound for the diffusion operator may take a much more general form.     

Towards a three-dimensional moving body incompressible flow solver with a linear deformable model

In this study, a three-dimensional fluid–structured parallelized solver is extended from the previous work (Niu et al., 2009 [1]) for moving body simulations. Based on the unified Eulerian and Lagrangian coordinate transformations, the unsteady three-dimensional incompressible Navier–Stokes equations with artificial compressibility (Chorin, 1967 [2]) in a dual-time stepping approach are first derived. To implement unsteady flow calculations, the dual-time stepping strategy including the LU decomposition method is used in the pseudo-time iteration and the second-order accurate backward difference is adopted to discretize the unsteady flow terms. Also, a third-order Roe type flux limited splitting is derived to evaluate the spatial difference of the convective fluxes. The original FORTRAN code is converted to the MPI code and tested on a 64-CPU IBM SP2. The parallel strategy here is based on the partitions of all do-loops in the original FORTRAN code and transferring the calculations inside the do-loop into different CPUs. The partition of the do-loop can be applied on the innermost loop, only or the last two inner loops depending on two-dimensional or three-dimensional problems. This kind of the parallel data partition of the loops is independent of what kind of the explicit or implicit type numerical algorithm used. Therefore, the current parallel approach can take advantage of the MPI language fully to transfer data efficiently among CPUs even for solving the governing equation implicitly. The test results show that a significant reduction of computing time in running the model and a near-linear speed up rate is achieved up to 32 CPUs at IBM SP2. The speed up rate is as high as 31 for using 64 IBM SP2 processors The test shows efficient parallel processing to provide prompt simulation of 3D cavity, unsteady dropping airfoil and blood flows in an aortic tube with a linear elastic modeling of wall motion is included here.     

Cooperative strategies of wireless access technologies: A game-theoretic analysis

The wireless Internet has to overcome the problem of spectrum scarcity as the number of mobile equipments could increase even by an order of magnitude in the next decade; the cooperation of mobile devices is foreseeable as a feasible solution to the problems. There exists a large body of literature on opportunistic ad hoc networking including Pelusi et al. (2006) [25], Chen et al. (2006) [26], Hui et al. (2005) [27]; however, the impact of the location of the devices on their access method selection is not yet appropriately dealt with. In this paper, we address this issue based on game-theoretic analyses. The key contribution of our work is threefold. First, we model the access method selection of mobile devices by extending the classical forwarding game with position, mobility, and availability of the devices. Second, we apply the model in game-theoretic analyses to better understand the optimal cooperation strategies in the presence of heterogeneous wireless technologies. We further extend our framework to include uncertainty. Finally, we present the applicability of the model in a cognitive radio scenario where complex structures of parameters are included.     

Mesoscale simulations of particulate flows with parallel distributed Lagrange multiplier technique

Fluid particulate flows are common phenomena in nature and industry. Modeling of such flows at micro and macro levels as well establishing relationships between these approaches are needed to understand properties of the particulate matter. We propose a computational technique based on the direct numerical simulation of the particulate flows. The numerical method is based on the distributed Lagrange multiplier technique following the ideas of Glowinski et al. [16] and Patankar [30]. Each particle is explicitly resolved on an Eulerian grid as a separate domain, using solid volume fractions. The fluid equations are solved through the entire computational domain, however, Lagrange multiplier constrains are applied inside the particle domain such that the fluid within any volume associated with a solid particle moves as an incompressible rigid body. Mutual forces for the fluid-particle interactions are internal to the system. Particles interact with the fluid via fluid dynamic equations, resulting in implicit fluid-rigid body coupling relations that produce realistic fluid flow around the particles (i.e., no-slip boundary conditions). The particle-particle interactions are implemented using explicit force-displacement interactions for frictional inelastic particles similar to the DEM method of Cundall et al. [10] with some modifications using a volume of an overlapping region as an input to the contact forces. The method is flexible enough to handle arbitrary particle shapes and size distributions. A parallel implementation of the method is based on the SAMRAI (Structured Adaptive Mesh Refinement Application Infrastructure) library, which allows handling of large amounts of rigid particles and enables local grid refinement. Accuracy and convergence of the presented method has been tested against known solutions for a falling particle as well as by examining fluid flows through stationary particle beds (periodic and cubic packing). To evaluate code performance and validate particle contact physics algorithm, we performed simulations of a representative experiment conducted at the U.C. Berkeley Thermal Hydraulic Lab for pebble flow through a narrow opening.     

Numerical resolution of discontinuous Galerkin methods for time dependent wave equations

The discontinuous Galerkin (DG) method is known to provide good wave resolution properties, especially for long time simulation. In this paper, using Fourier analysis, we provide a quantitative error analysis for the semi-discrete DG method applied to time dependent linear convection equations with periodic boundary conditions. We apply the same technique to show that the error is of order k + 2 superconvergent at Radau points on each element and of order 2k + 1 superconvergent at the downwind point of each element, when using piecewise polynomials of degree k. An analysis of the fully discretized approximation is also provided. We compute the number of points per wavelength required to obtain a fixed error for several fully discrete schemes. Numerical results are provided to verify our error analysis.     

A rational high-order compact ADI method for unsteady convection–diffusion equations

Based on a fourth-order compact difference formula for the spatial discretization, which is currently proposed for the one-dimensional (1D) steady convection–diffusion problem, and the Crank–Nicolson scheme for the time discretization, a rational high-order compact alternating direction implicit (ADI) method is developed for solving two-dimensional (2D) unsteady convection–diffusion problems. The method is unconditionally stable and second-order accurate in time and fourth-order accurate in space. The resulting scheme in each ADI computation step corresponds to a tridiagonal matrix equation which can be solved by the application of the 1D tridiagonal Thomas algorithm with a considerable saving in computing time. Three examples supporting our theoretical analysis are numerically solved. The present method not only shows higher accuracy and better phase and amplitude error properties than the standard second-order Peaceman–Rachford ADI method in Peaceman and Rachford (1959) [4], the fourth-order ADI method of Karaa and Zhang (2004) [5] and the fourth-order ADI method of Tian and Ge (2007) [23], but also proves more effective than the fourth-order Padé ADI method of You (2006) [6], in the aspect of computational cost. The method proposed for the diffusion–convection problems is easy to implement and can also be used to solve pure diffusion or pure convection problems.     

Using Hopfield neural network to optimize fuel rod loading patterns in VVER/1000 reactor by applying axial variation of enrichment distribution

The present work investigates an appropriate way to solve the problem of optimizing fuel management in a VVER/1000 reactor. To automate this procedure, a computer program has been developed. This program suggests an optimal core configuration which is determined according to established safety constraints. The suggested solution is based on the use of coupled programs, one of which is the nuclear code, for making a database and modeling the core, and another one is the Hopfield neural network. In addition to we applied axial variations of enrichment in fuel rods to flat the flux core as novel role. This computational procedure consists of three main steps. The first one consists of creating the cross section database and calculating neutronic parameters by using WIMSD4 and CITATION codes. The second one consists of finding the best axial variations distributions of enrichment to create a fuel rod pattern by using Hopfield neural network artificial (HNNA) and the cross section database. The third one consists of loading of the fuel rods by the suggested fuel rod patterns and finding the optimum core configuration by HNNA that based on minimizing power peaking factor (PPF, PPF = maximum power/average power) and maximizing the effective multiplication factor (k_eff, the ratio of the number of neutrons in two successive fission generations). The procedure uses the optimized parameters in order to find configurations in which k_eff is maximized. The penalty function is applied to limit the value of local PPF in the neighborhood fuel assemblies. Therefore, in this paper, we proposed a new approach for the use of Hopfield neural network to guide the heuristic search, and applied axial variations distributions of enrichment as novel method to flat the neutron flux and for evaluating the obtained results pertaining to the first core.The results show that applying the HNNA led us to the appropriate PPF and k_eff. Also, applying HNNA and axial variation of enrichment is promising to reach the flattening neutronic flux and guaranteeing safety condition in the reactor core. Therefore, we achieved to a set of two basic parameters PPF and k_eff as effective factors on satisfying the safety constraints of VVER/1000 reactor core.     

SPH truncation error in estimating a 3D function

The SPH truncation error (ε_T) can be defined as the sum of the integral kernel and the particle approximation error in Smoothed Particle Hydrodynamics modelling. Following the procedure proposed by Quinlan et al. [16] for a 1D generic derivative, we have derived an approximated 3D formulation of ε_T in reproducing a generic function. This kind of estimation is implemented in some SPH models in order to reproduce density or some transported scalars. Then a corresponding sensitivity analysis of ε_T has been performed adopting regular and irregular distributions of particles, arranged within a cube, delimited by lateral walls at each side. The evolution of ε_T has been analyzed and compared to the proposed formulation, which has been numerically estimated under some simple conditions. The SPH truncation error has then been investigated on a simple free surface test case: a supercritical flow over a channel sill. We have developed some conclusions about the dependence of ε_T on the position of the particles (inner or boundary), the shape of the function to be reproduced (f), the kernel support size (h), the particle volumes (ω), the kernel function (W), a non-dimensional distance between the volume barycentre and the particle location (δ), and a geometric anisotropy index of the particle volumes (I). We have finally underlined the difference between non-consistent simulations and estimations using Shepard’s correction.     

A statistical model of fracture for a 2D hexagonal mesh: The Cell Network Model of Fracture for the bamboo Guadua angustifolia

A 2D, hexagonal in geometry, statistical model of fracture is proposed. The model is based on the drying fracture process of the bamboo Guadua angustifolia. A network of flexible cells are joined by brittle junctures of fixed Young moduli that break at a certain thresholds in tensile force. The system is solved by means of the Finite Element Method (FEM). The distribution of avalanche breakings exhibits a power law with exponent −2.93(9), in agreement with the random fuse model (Bhattacharyya and Chakrabarti, 2006) [1].     

Improved parallelized hybrid DSMC–NS method

In this paper, an improved parallelized hybrid DSMC–NS (Navier–Stokes method) algorithm, as compared to the previous work [1], is presented. A detailed kinetic velocity sampling study is conducted with a two-dimensional supersonic flow (M_∞ = 4) past a 25° finite wedge. It shows most of the boundary layer region is in nearly thermal equilibrium, even with very high continuum breakdown parameter based on velocity, velocity gradient and local mean free path. A new continuum breakdown parameter based on pressure is designed to effectively “exclude” the “false” breakdown region such as the boundary layer. An improved hybrid DSMC–NS algorithm is verified using the same wedge flow case. Results show that the improved algorithm can greatly reduce the computational cost while maintaining essentially the same accuracy. A hypersonic flow (M_∞ = 12) past a square cylinder is also employed to exhibit the capability of the improved hybrid DSMC–NS method.     

Cooperative and competitive concurrency in scientific computing. A full open-source upgrade of the program for dynamical calculations of RHEED intensity oscillations

A computational model is a computer program, which attempts to simulate an abstract model of a particular system. Computational models use enormous calculations and often require supercomputer speed. As personal computers are becoming more and more powerful, more laboratory experiments can be converted into computer models that can be interactively examined by scientists and students without the risk and cost of the actual experiments. The future of programming is concurrent programming. The threaded programming model provides application programmers with a useful abstraction of concurrent execution of multiple tasks. The objective of this release is to address the design of architecture for scientific application, which may execute as multiple threads execution, as well as implementations of the related shared data structures.

New version program summary

Program title: GrowthCPCatalogue identifier: ADVL_v4_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADVL_v4_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.: 32 269No. of bytes in distributed program, including test data, etc.: 8 234 229Distribution format: tar.gzProgramming language: Free Object PascalComputer: multi-core x64-based PCOperating system: Windows XP, Vista, 7Has the code been vectorised or parallelized?: NoRAM: More than 1 GB. The program requires a 32-bit or 64-bit processor to run the generated code. Memory is addressed using 32-bit (on 32-bit processors) or 64-bit (on 64-bit processors with 64-bit addressing) pointers. The amount of addressed memory is limited only by the available amount of virtual memory.Supplementary material: The figures mentioned in the “Summary of revisions” section can be obtained here.Classification: 4.3, 7.2, 6.2, 8, 14External routines: Lazarus [1]Catalogue identifier of previous version: ADVL_v3_0Journal reference of previous version: Comput. Phys. Comm. 181 (2010) 709Does the new version supersede the previous version?: YesNature of problem: Reflection high-energy electron diffraction (RHEED) is an important in-situ analysis technique, which is capable of giving quantitative information about the growth process of thin layers and its control. It can be used to calibrate growth rate, analyze surface morphology, calibrate surface temperature, monitor the arrangement of the surface atoms, and provide information about growth kinetics. Such control allows the development of structures where the electrons can be confined in space, giving quantum wells or even quantum dots. In order to determine the atomic positions of atoms in the first few layers, the RHEED intensity must be measured as a function of the scattering angles and then compared with dynamic calculations. The objective of this release is to address the design of architecture for application that simulates the rocking curves RHEED intensities during hetero-epitaxial growth process of thin films.Solution method: The GrowthCP is a complex numerical model that uses multiple threads for simulation of epitaxial growth of thin layers. This model consists of two transactional parts. The first part is a mathematical model being based on the Runge–Kutta method with adaptive step-size control. The second part represents first-principles of the one-dimensional RHEED computational model. This model is based on solving a one-dimensional Schrödinger equation. Several problems can arise when applications contain a mixture of data access code, numerical code, and presentation code. Such applications are difficult to maintain, because interdependencies between all the components cause strong ripple effects whenever a change is made anywhere. Adding new data views often requires reimplementing a numerical code, which then requires maintenance in multiple places. In order to solve problems of this type, the computational and threading layers of the project have been implemented in the form of one design pattern as a part of Model-View-Controller architecture.Reasons for new version: Responding to the users? feedback the Growth09 project has been upgraded to a standard that allows the carrying out of sample computations of the RHEED intensities for a disordered surface for a wide range of single- and epitaxial hetero-structures. The design pattern on which the project is based has also been improved. It is shown that this model can be effectively used for multithreaded growth simulations of thin epitaxial layers and corresponding RHEED intensities for a wide range of single- and hetero-structures. Responding to the users? feedback the present release has been implemented using a well-documented free compiler [1] not requiring the special configuration and installation additional libraries.Summary of revisions:

• 1. 

  The logical structure of the Growth09 program has been modified according to the scheme showed in Fig. 1.1 The class diagram in Fig. 11 is a static view of the main platform-specific elements of the GrowthCP architecture. Fig. 21 provides a dynamic view by showing the creation and destruction simplistic sequence diagram for the process.

• 2. 

  The program requires the user to provide the appropriate parameters in the form of a knowledge base for the crystal structures under investigation. These parameters are loaded from the parameters.ini files at run-time. Instructions to prepare the .ini files can be found in the new distribution.

• 3. 

  The program enables carrying out different growth models and one-dimensional dynamical RHEED calculations for the fcc lattice with basis of three-atoms, fcc lattice with basis of two-atoms, fcc lattice with single atom basis, Zinc-Blende, Sodium Chloride, and Wurtzite crystalline structures and hetero-structures, but yet the Fourier component of the scattering potential in the TRHEEDCalculations.crystPotUgXXX() procedure can be modified and implemented according to users? specific application requirements. The Fourier component of the scattering potential of the whole crystalline hetero-structures can be determined as a sum of contributions coming from all thin slices of individual atomic layers. To carry out one-dimensional calculations of the scattering potentials, the program uses properly constructed self-consistent procedures.

• 4. 

  Each component of the system shown in Figs. 11 and 21 is fully extendable and can easily be adapted to new changeable requirements. Two essential logical elements of the system, i.e. TGrowthTransaction and TRHEEDCalculations classes, were designed and implemented in this way for them to pass the information to themselves without the need to use the data-exchange files given. In consequence each of them can be independently modified and/or extended. Implementing other types of differential equations and the different algorithm for solving them in the TGrowthTransaction class does not require another implementation of the TRHEEDCalculations class. Similarly, implementing other forms of scattering potential and different algorithm for RHEED calculation stays without the influence on the TGrowthTransaction class construction.

Unusual features: The program is distributed in the form of main project GrowthCP.lpr, with associated files, and should be compiled using Lazarus IDE. The program should be compiled with English/USA regional and language options.Running time: The typical running time is machine and user-parameters dependent.References:

• [1] 

  http://sourceforge.net/projects/lazarus/files/.

     

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.