Numerical approximation of the Ginzburg-Landau equation with memory effects in the dynamics of phase transitions

We consider the out-of-equilibrium time evolution of a nonconserved order parameter using the Ginzburg-Landau equation including memory effects. Memory effects are expected to play important role on the nonequilibrium dynamics for fast phase transitions, in which the time scales of the rapid phase conversion are comparable to the microscopic time scales. We consider two numerical approximation schemes based on Fourier collocation and finite difference methods and perform a numerical analysis with respect to the existence, stability and convergence of the schemes. We present results of direct numerical simulations for one and three spatial dimensions, and examine numerically the stability and convergence of both approximation schemes.     

KANTBP 2.0: New version of a program for computing energy levels, reaction matrix and radial wave functions in the coupled-channel hyperspherical adiabatic approach

A FORTRAN 77 program for calculating energy values, reaction matrix and corresponding radial wave functions in a coupled-channel approximation of the hyperspherical adiabatic approach is presented. In this approach, a multi-dimensional Schrödinger equation is reduced to a system of the coupled second-order ordinary differential equations on a finite interval with homogeneous boundary conditions: (i) the Dirichlet, Neumann and third type at the left and right boundary points for continuous spectrum problem, (ii) the Dirichlet and Neumann type conditions at left boundary point and Dirichlet, Neumann and third type at the right boundary point for the discrete spectrum problem. The resulting system of radial equations containing the potential matrix elements and first-derivative coupling terms is solved using high-order accuracy approximations of the finite element method. As a test desk, the program is applied to the calculation of the reaction matrix and radial wave functions for 3D-model of a hydrogen-like atom in a homogeneous magnetic field. This version extends the previous version 1.0 of the KANTBP program [O. Chuluunbaatar, A.A. Gusev, A.G. Abrashkevich, A. Amaya-Tapia, M.S. Kaschiev, S.Y. Larsen, S.I. Vinitsky, Comput. Phys. Commun. 177 (2007) 649-675].

Program summary

Program title: KANTBPCatalogue identifier: ADZH_v2_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADZH_v2_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.: 20 403No. of bytes in distributed program, including test data, etc.: 147 563Distribution format: tar.gzProgramming language: FORTRAN 77Computer: Intel Xeon EM64T, Alpha 21264A, AMD Athlon MP, Pentium IV Xeon, Opteron 248, Intel Pentium IVOperating system: OC Linux, Unix AIX 5.3, SunOS 5.8, Solaris, Windows XPRAM: This depends on

1.

the number of differential equations;

2.

the number and order of finite elements;

3.

the number of hyperradial points; and

4.

the number of eigensolutions required.

The test run requires 2 MBClassification: 2.1, 2.4External routines: GAULEG and GAUSSJ [2]Nature of problem: In the hyperspherical adiabatic approach [3-5], a multidimensional Schrödinger equation for a two-electron system [6] or a hydrogen atom in magnetic field [7-9] is reduced by separating radial coordinate ρ from the angular variables to a system of the second-order ordinary differential equations containing the potential matrix elements and first-derivative coupling terms. The purpose of this paper is to present the finite element method procedure based on the use of high-order accuracy approximations for calculating approximate eigensolutions of the continuum spectrum for such systems of coupled differential equations on finite intervals of the radial variable ρ∈[ρmin,ρmax]. This approach can be used in the calculations of effects of electron screening on low-energy fusion cross sections [10-12].Solution method: The boundary problems for the coupled second-order differential equations are solved by the finite element method using high-order accuracy approximations [13]. The generalized algebraic eigenvalue problem AF=EBF with respect to pair unknowns (E,F) arising after the replacement of the differential problem by the finite-element approximation is solved by the subspace iteration method using the SSPACE program [14]. The generalized algebraic eigenvalue problem (A−EB)F=λDF with respect to pair unknowns (λ,F) arising after the corresponding replacement of the scattering boundary problem in open channels at fixed energy value, E, is solved by the LDLT factorization of symmetric matrix and back-substitution methods using the DECOMP and REDBAK programs, respectively [14]. As a test desk, the program is applied to the calculation of the reaction matrix and corresponding radial wave functions for 3D-model of a hydrogen-like atom in a homogeneous magnetic field described in [9] on finite intervals of the radial variable ρ∈[ρmin,ρmax]. For this benchmark model the required analytical expressions for asymptotics of the potential matrix elements and first-derivative coupling terms, and also asymptotics of radial solutions of the boundary problems for coupled differential equations have been produced with help of a MAPLE computer algebra system.Restrictions: The computer memory requirements depend on:

1.

the number of differential equations;

2.

the number and order of finite elements;

3.

the total number of hyperradial points; and

4.

the number of eigensolutions required.

Restrictions due to dimension sizes may be easily alleviated by altering PARAMETER statements (see Section 3 and [1] for details). The user must also supply subroutine POTCAL for evaluating potential matrix elements. The user should also supply subroutines ASYMEV (when solving the eigenvalue problem) or ASYMS0 and ASYMSC (when solving the scattering problem) which evaluate asymptotics of the radial wave functions at left and right boundary points in case of a boundary condition of the third type for the above problems.Running time: The running time depends critically upon:

1.

the number of differential equations;

2.

the number and order of finite elements;

3.

the total number of hyperradial points on interval [ρmin,ρmax]; and

4.

the number of eigensolutions required.

The test run which accompanies this paper took 2 s without calculation of matrix potentials on the Intel Pentium IV 2.4 GHz.References:[1] O. Chuluunbaatar, A.A. Gusev, A.G. Abrashkevich, A. Amaya-Tapia, M.S. Kaschiev, S.Y. Larsen, S.I. Vinitsky, Comput. Phys. Commun. 177 (2007) 649-675; http://cpc.cs.qub.ac.uk/summaries/ADZHv10.html.[2] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, Cambridge, 1986.[3] J. Macek, J. Phys. B 1 (1968) 831-843.[4] U. Fano, Rep. Progr. Phys. 46 (1983) 97-165.[5] C.D. Lin, Adv. Atom. Mol. Phys. 22 (1986) 77-142.[6] A.G. Abrashkevich, D.G. Abrashkevich, M. Shapiro, Comput. Phys. Commun. 90 (1995) 311-339.[7] M.G. Dimova, M.S. Kaschiev, S.I. Vinitsky, J. Phys. B 38 (2005) 2337-2352.[8] O. Chuluunbaatar, A.A. Gusev, V.L. Derbov, M.S. Kaschiev, L.A. Melnikov, V.V. Serov, S.I. Vinitsky, J. Phys. A 40 (2007) 11485-11524.[9] O. Chuluunbaatar, A.A. Gusev, V.P. Gerdt, V.A. Rostovtsev, S.I. Vinitsky, A.G. Abrashkevich, M.S. Kaschiev, V.V. Serov, Comput. Phys. Commun. 178 (2007) 301 330; http://cpc.cs.qub.ac.uk/summaries/AEAAv10.html.[10] H.J. Assenbaum, K. Langanke, C. Rolfs, Z. Phys. A 327 (1987) 461-468.[11] V. Melezhik, Nucl. Phys. A 550 (1992) 223-234.[12] L. Bracci, G. Fiorentini, V.S. Melezhik, G. Mezzorani, P. Pasini, Phys. Lett. A 153 (1991) 456-460.[13] A.G. Abrashkevich, D.G. Abrashkevich, M.S. Kaschiev, I.V. Puzynin, Comput. Phys. Commun. 85 (1995) 40-64.[14] K.J. Bathe, Finite Element Procedures in Engineering Analysis, Englewood Cliffs, Prentice-Hall, New York, 1982.     

A numerical differentiation library exploiting parallel architectures

We present a software library for numerically estimating first and second order partial derivatives of a function by finite differencing. Various truncation schemes are offered resulting in corresponding formulas that are accurate to order O(h), O(h²), and O(h⁴), h being the differencing step. The derivatives are calculated via forward, backward and central differences. Care has been taken that only feasible points are used in the case where bound constraints are imposed on the variables. The Hessian may be approximated either from function or from gradient values. There are three versions of the software: a sequential version, an OpenMP version for shared memory architectures and an MPI version for distributed systems (clusters). The parallel versions exploit the multiprocessing capability offered by computer clusters, as well as modern multi-core systems and due to the independent character of the derivative computation, the speedup scales almost linearly with the number of available processors/cores.

Program summary

Program title: NDL (Numerical Differentiation Library)Catalogue identifier: AEDG_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEDG_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.: 73 030No. of bytes in distributed program, including test data, etc.: 630 876Distribution format: tar.gzProgramming language: ANSI FORTRAN-77, ANSI C, MPI, OPENMPComputer: Distributed systems (clusters), shared memory systemsOperating system: Linux, SolarisHas the code been vectorised or parallelized?: YesRAM: The library uses O(N) internal storage, N being the dimension of the problemClassification: 4.9, 4.14, 6.5Nature of problem: The numerical estimation of derivatives at several accuracy levels is a common requirement in many computational tasks, such as optimization, solution of nonlinear systems, etc. The parallel implementation that exploits systems with multiple CPUs is very important for large scale and computationally expensive problems.Solution method: Finite differencing is used with carefully chosen step that minimizes the sum of the truncation and round-off errors. The parallel versions employ both OpenMP and MPI libraries.Restrictions: The library uses only double precision arithmetic.Unusual features: The software takes into account bound constraints, in the sense that only feasible points are used to evaluate the derivatives, and given the level of the desired accuracy, the proper formula is automatically employed.Running time: Running time depends on the function's complexity. The test run took 15 ms for the serial distribution, 0.6 s for the OpenMP and 4.2 s for the MPI parallel distribution on 2 processors.     

Numerical modeling of two-phase transonic flow

Our work is aimed at the development of numerical method for the modeling of transonic flow of wet steam including condensation/evaporation phase change. We solve a system of PDE’s consisting of Euler or Navier-Stokes equations for the mixture of vapor and liquid droplets and transport equations for the integral parameters describing the droplet size spectra. Numerical method is based on a fractional step technique due to the stiff character of source terms, i.e. we solve separately the set of homogenous PDE’s by the finite volume method and the remaining set of ODE’s either by explicit Runge-Kutta or implicit Euler method. The finite volume method is based on the Lax-Wendroff scheme with conservative artificial dissipation terms for structured grid. We also note result achieved by recently developed finite volume method with VFFC scheme. We discuss numerical results of steady and unsteady two-phase transonic flow in 2D nozzle, 2D and 3D turbine cascade and 2D turbine stage with moving rotor cascade.     

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.     

GDF v2.0, an enhanced version of GDF

An improved version of the function estimation program GDF is presented. The main enhancements of the new version include: multi-output function estimation, capability of defining custom functions in the grammar and selection of the error function. The new version has been evaluated on a series of classification and regression datasets, that are widely used for the evaluation of such methods. It is compared to two known neural networks and outperforms them in 5 (out of 10) datasets.

Program summary

Title of program: GDF v2.0Catalogue identifier: ADXC_v2_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADXC_v2_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.: 98 147No. of bytes in distributed program, including test data, etc.: 2 040 684Distribution format: tar.gzProgramming language: GNU C++Computer: The program is designed to be portable in all systems running the GNU C++ compilerOperating system: Linux, Solaris, FreeBSDRAM: 200000 bytesClassification: 4.9Does the new version supersede the previous version?: YesNature of problem: The technique of function estimation tries to discover from a series of input data a functional form that best describes them. This can be performed with the use of parametric models, whose parameters can adapt according to the input data.Solution method: Functional forms are being created by genetic programming which are approximations for the symbolic regression problem.Reasons for new version: The GDF package was extended in order to be more flexible and user customizable than the old package. The user can extend the package by defining his own error functions and he can extend the grammar of the package by adding new functions to the function repertoire. Also, the new version can perform function estimation of multi-output functions and it can be used for classification problems.Summary of revisions: The following features have been added to the package GDF:

•

Multi-output function approximation. The package can now approximate any function . This feature gives also to the package the capability of performing classification and not only regression.

•

User defined function can be added to the repertoire of the grammar, extending the regression capabilities of the package. This feature is limited to 3 functions, but easily this number can be increased.

•

Capability of selecting the error function. The package offers now to the user apart from the mean square error other error functions such as: mean absolute square error, maximum square error. Also, user defined error functions can be added to the set of error functions.

•

More verbose output. The main program displays more information to the user as well as the default values for the parameters. Also, the package gives to the user the capability to define an output file, where the output of the gdf program for the testing set will be stored after the termination of the process.

Additional comments: A technical report describing the revisions, experiments and test runs is packaged with the source code.Running time: Depending on the train data.     

ODPEVP: A program for computing eigenvalues and eigenfunctions and their first derivatives with respect to the parameter of the parametric self-adjoined Sturm-Liouville problem

A FORTRAN 77 program is presented for calculating with the given accuracy eigenvalues, eigenfunctions and their first derivatives with respect to the parameter of the parametric self-adjoined Sturm-Liouville problem with the parametric third type boundary conditions on the finite interval. The program calculates also potential matrix elements - integrals of the eigenfunctions multiplied by their first derivatives with respect to the parameter. Eigenvalues and matrix elements computed by the ODPEVP program can be used for solving the bound state and multi-channel scattering problems for a system of the coupled second-order ordinary differential equations with the help of the KANTBP programs [O. Chuluunbaatar, A.A. Gusev, A.G. Abrashkevich, A. Amaya-Tapia, M.S. Kaschiev, S.Y. Larsen, S.I. Vinitsky, Comput. Phys. Commun. 177 (2007) 649-675; O. Chuluunbaatar, A.A. Gusev, S.I. Vinitsky, A.G. Abrashkevich, Comput. Phys. Commun. 179 (2008) 685-693]. As a test desk, the program is applied to the calculation of the potential matrix elements for an integrable 2D-model of three identical particles on a line with pair zero-range potentials, a 3D-model of a hydrogen atom in a homogeneous magnetic field and a hydrogen atom on a three-dimensional sphere.

Program summary

Program title: ODPEVPCatalogue identifier: AEDV_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEDV_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC license, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 3001No. of bytes in distributed program, including test data, etc.: 24 195Distribution format: tar.gzProgramming language: FORTRAN 77Computer: Intel Xeon EM64T, Alpha 21264A, AMD Athlon MP, Pentium IV Xeon, Opteron 248, Intel Pentium IVOperating system: OC Linux, Unix AIX 5.3, SunOS 5.8, Solaris, Windows XPRAM: depends on

1.

the number and order of finite elements;

2.

the number of points; and

3.

the number of eigenfunctions required.

Test run requires 4 MBClassification: 2.1, 2.4External routines: GAULEG [3]Nature of problem: The three-dimensional boundary problem for the elliptic partial differential equation with an axial symmetry similar to the Schrödinger equation with the Coulomb and transverse oscillator potentials is reduced to the two-dimensional one. The latter finds wide applications in modeling of photoionization and recombination of oppositively charged particles (positrons, antiprotons) in the magnet-optical trap [4], optical absorption in quantum wells [5], and channeling of likely charged particles in thin doped films [6,7] or neutral atoms and molecules in artificial waveguides or surfaces [8,9]. In the adiabatic approach [10] known in mathematics as Kantorovich method [11] the solution of the two-dimensional elliptic partial differential equation is expanded over basis functions with respect to the fast variable (for example, angular variable) and depended on the slow variable (for example, radial coordinate ) as a parameter. An averaging of the problem by such a basis leads to a system of the second-order ordinary differential equations which contain potential matrix elements and the first-derivative coupling terms (see, e.g., [12,13,14]). The purpose of this paper is to present the finite element method procedure based on the use of high-order accuracy approximations for calculating eigenvalues, eigenfunctions and their first derivatives with respect to the parameter of the parametric self-adjoined Sturm-Liouville problem with the parametric third type boundary conditions on the finite interval. The program developed calculates potential matrix elements - integrals of the eigenfunctions multiplied by their derivatives with respect to the parameter. These matrix elements can be used for solving the bound state and multi-channel scattering problems for a system of the coupled second-order ordinary differential equations with the help of the KANTBP programs [1,2].Solution method: The parametric self-adjoined Sturm-Liouville problem with the parametric third type boundary conditions is solved by the finite element method using high-order accuracy approximations [15]. The generalized algebraic eigenvalue problem AF=EBF with respect to a pair of unknown (E,F) arising after the replacement of the differential problem by the finite-element approximation is solved by the subspace iteration method using the SSPACE program [16]. First derivatives of the eigenfunctions with respect to the parameter which contained in potential matrix elements of the coupled system equations are obtained by solving the inhomogeneous algebraic equations. As a test desk, the program is applied to the calculation of the potential matrix elements for an integrable 2D-model of three identical particles on a line with pair zero-range potentials described in [1,17,18], a 3D-model of a hydrogen atom in a homogeneous magnetic field described in [14,19] and a hydrogen atom on a three-dimensional sphere [20].Restrictions: The computer memory requirements depend on:

1.

the number and order of finite elements;

2.

the number of points; and

3.

the number of eigenfunctions required.

Restrictions due to dimension sizes may be easily alleviated by altering PARAMETER statements (see sections below and listing for details). The user must also supply DOUBLE PRECISION functions POTCCL and POTCC1 for evaluating potential function U(ρ,z) of Eq. (1) and its first derivative with respect to parameter ρ. The user should supply DOUBLE PRECISION functions F1FUNC and F2FUNC that evaluate functions f₁(z) and f₂(z) of Eq. (1). The user must also supply subroutine BOUNCF for evaluating the parametric third type boundary conditions.Running time: The running time depends critically upon:

1.

the number and order of finite elements;

2.

the number of points on interval [zmin,zmax]; and

3.

the number of eigenfunctions required.

The test run which accompanies this paper took 2 s with calculation of matrix potentials on the Intel Pentium IV 2.4 GHz.References:

[1]

O. Chuluunbaatar, A.A. Gusev, A.G. Abrashkevich, A. Amaya-Tapia, M.S. Kaschiev, S.Y. Larsen, S.I. Vinitsky, Comput. Phys. Comm. 177 (2007) 649-675

[2]

O. Chuluunbaatar, A.A. Gusev, S.I. Vinitsky, A.G. Abrashkevich, Comput. Phys. Comm. 179 (2008) 685-693.

[3]

W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, Cambridge, 1986.

[4]

O. Chuluunbaatar, A.A. Gusev, S.I. Vinitsky, V.L. Derbov, L.A. Melnikov, V.V. Serov, Phys. Rev. A 77 (2008) 034702-1-4.

[5]

E.M. Kazaryan, A.A. Kostanyan, H.A. Sarkisyan, Physica E 28 (2005) 423-430.

[6]

Yu.N. Demkov, J.D. Meyer, Eur. Phys. J. B 42 (2004) 361-365.

[7]

P.M. Krassovitskiy, N.Zh. Takibaev, Bull. Russian Acad. Sci. Phys. 70 (2006) 815-818.

[8]

V.S. Melezhik, J.I. Kim, P. Schmelcher, Phys. Rev. A 76 (2007) 053611-1-15.

[9]

F.M. Pen'kov, Phys. Rev. A 62 (2000) 044701-1-4.

[10]

M. Born, X. Huang, Dynamical Theory of Crystal Lattices, The Clarendon Press, Oxford, England, 1954.

[11]

L.V. Kantorovich, V.I. Krylov, Approximate Methods of Higher Analysis, Wiley, New York, 1964.

[12]

U. Fano, Colloq. Int. C.N.R.S. 273 (1977) 127;

A.F. Starace, G.L. Webster, Phys. Rev. A 19 (1979) 1629-1640.

[13]

C.V. Clark, K.T. Lu, A.F. Starace, in: H.G. Beyer, H. Kleinpoppen (eds.), Progress in Atomic Spectroscopy, Part C, Plenum, New York, 1984, pp. 247-320.

[14]

O. Chuluunbaatar, A.A. Gusev, V.L. Derbov, M.S. Kaschiev, L.A. Melnikov, V.V. Serov, S.I. Vinitsky, J. Phys. A 40 (2007) 11485-11524.

[15]

A.G. Abrashkevich, D.G. Abrashkevich, M.S. Kaschiev, I.V. Puzynin, Comput. Phys. Comm. 85 (1995) 40-64.

[16]

K.J. Bathe, Finite Element Procedures in Engineering Analysis, Englewood Cliffs, Prentice-Hall, New York, 1982.

[17]

O. Chuluunbaatar, A.A. Gusev, M.S. Kaschiev, V.A. Kaschieva, A. Amaya-Tapia, S.Y. Larsen, S.I. Vinitsky, J. Phys. B 39 (2006) 243-269.

[18]

Yu.A. Kuperin, P.B. Kurasov, Yu.B. Melnikov, S.P. Merkuriev, Ann. Phys. 205 (1991) 330-361.

[19]

O. Chuluunbaatar, A.A. Gusev, V.P. Gerdt, V.A. Rostovtsev, S.I. Vinitsky, A.G. Abrashkevich, M.S. Kaschiev, V.V. Serov, Comput. Phys. Comm. 178 (2008) 301-330.

[20]

A.G. Abrashkevich, M.S. Kaschiev, S.I. Vinitsky, J. Comp. Phys. 163 (2000) 328-348.

     

An application of the interpolating scaling functions to wave packet propagation

The wave packet propagation in the basis of interpolating scaling functions (ISF) is studied. The ISF are well known in the multiresolution analysis based on spline biorthogonal wavelets. The ISF form a cardinal basis set corresponding to an equidistantly spaced grid. They have compact support of the size determined by the order of the underlying interpolating polynomial. In this basis the potential energy matrix is diagonal. The kinetic energy matrix is sparse, and in the 1D case, has a band-diagonal structure. An important future of the basis is that matrix elements of a Hamiltonian are exactly computed by means of simple algebraic transformations efficiently implemented numerically. Therefore, the number of grid points and the order of the underlying interpolating polynomial can easily be varied allowing one to approach the accuracy of pseudospectral methods in a regular manner, similar to the high order finite difference methods. The results for the calculation of the H+H₂ collinear collision shows that the ISF provide one with an accurate and efficient representation for use in wave packet propagation method.     

Static and dynamic glass transitions in the 10-state Potts glass: What can Monte Carlo simulations contribute?

The p-state Potts glass with infinite range Gaussian interactions can be solved exactly in the thermodynamic limit and exhibits an unconventional phase behavior if p>4: A dynamical transition from ergodic to non-ergodic behavior at a temperature T_D is followed by a first order transition at T₀<T_D, where a glass order parameter appears discontinuously, although the latent heat is zero. If one assumes that a similar scenario occurs for the structural glass transition as well (though with the singular behavior at T_D rounded off), the p-state Potts glass should be a good test case to develop methods to deal with finite size effects for the static as well as the dynamic transition, and to check what remnants of these unconventional transitions are left in finite sized systems, as they are used in simulations. While it is shown that a sensible extrapolation N→∞ of the simulation results are compatible with the exact results, we find that it would be rather difficult to obtain a correct understanding of the behavior of the system in the thermodynamic limit if only the numerical data would be available.     

MinFinder v2.0: An improved version of MinFinder

A new version of the “MinFinder” program is presented that offers an augmented linking procedure for Fortran-77 subprograms, two additional stopping rules and a new start-point rejection mechanism that saves a significant portion of gradient and function evaluations. The method is applied on a set of standard test functions and the results are reported.

New version program summary

Program title: MinFinder v2.0Catalogue identifier: ADWU_v2_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADWU_v2_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.: 14 150No. of bytes in distributed program, including test data, etc.: 218 144Distribution format: tar.gzProgramming language used: GNU C++, GNU FORTRAN, GNU CComputer: The program is designed to be portable in all systems running the GNU C++ compilerOperating system: Linux, Solaris, FreeBSDRAM: 200 000 bytesClassification: 4.9Catalogue identifier of previous version: ADWU_v1_0Journal reference of previous version: Computer Physics Communications 174 (2006) 166-179Does the new version supersede the previous version?: YesNature of problem: A multitude of problems in science and engineering are often reduced to minimizing a function of many variables. There are instances that a local optimum does not correspond to the desired physical solution and hence the search for a better solution is required. Local optimization techniques can be trapped in any local minimum. Global optimization is then the appropriate tool. For example, solving a non-linear system of equations via optimization, one may encounter many local minima that do not correspond to solutions, i.e. they are far from zero.Solution method: Using a uniform pdf, points are sampled from a rectangular domain. A clustering technique, based on a typical distance and a gradient criterion, is used to decide from which points a local search should be started. Further searching is terminated when all the local minima inside the search domain are thought to be found. This is accomplished via three stopping rules: the “double-box” stopping rule, the “observables” stopping rule and the “expected minimizers” stopping rule.Reasons for the new version: The link procedure for source code in Fortran 77 is enhanced, two additional stopping rules are implemented and a new criterion for accepting-start points, that economizes on function and gradient calls, is introduced.Summary of revisions:

1.

Addition of command line parameters to the utility program make_program.

2.

Augmentation of the link process for Fortran 77 subprograms, by linking the final executable with the g2c library.

3.

Addition of two probabilistic stopping rules.

4.

Introduction of a rejection mechanism to the Checking step of the original method, that reduces the number of gradient evaluations.

Additional comments: A technical report describing the revisions, experiments and test runs is packaged with the source code.Running time: Depending on the objective function.     

JADAMILU: a software code for computing selected eigenvalues of large sparse symmetric matrices

A new software code for computing selected eigenvalues and associated eigenvectors of a real symmetric matrix is described. The eigenvalues are either the smallest or those closest to some specified target, which may be in the interior of the spectrum. The underlying algorithm combines the Jacobi-Davidson method with efficient multilevel incomplete LU (ILU) preconditioning. Key features are modest memory requirements and robust convergence to accurate solutions. Parameters needed for incomplete LU preconditioning are automatically computed and may be updated at run time depending on the convergence pattern. The software is easy to use by non-experts and its top level routines are written in FORTRAN 77. Its potentialities are demonstrated on a few applications taken from computational physics.

Program summary

Program title: JADAMILUCatalogue identifier: ADZT_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADZT_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.: 101 359No. of bytes in distributed program, including test data, etc.: 7 493 144Distribution format: tar.gzProgramming language: Fortran 77Computer: Intel or AMD with g77 and pgf; Intel EM64T or Itanium with ifort; AMD Opteron with g77, pgf and ifort; Power (IBM) with xlf90.Operating system: Linux, AIXRAM: problem dependentWord size: real:8; integer: 4 or 8, according to user's choiceClassification: 4.8Nature of problem: Any physical problem requiring the computation of a few eigenvalues of a symmetric matrix.Solution method: Jacobi-Davidson combined with multilevel ILU preconditioning.Additional comments: We supply binaries rather than source code because JADAMILU uses the following external packages:

•

MC64. This software is copyrighted software and not freely available. COPYRIGHT (c) 1999 Council for the Central Laboratory of the Research Councils.

•

AMD. Copyright (c) 2004-2006 by Timothy A. Davis, Patrick R. Amestoy, and Iain S. Duff. All Rights Reserved. Source code is distributed by the authors under the GNU LGPL licence.

•

BLAS. The reference BLAS is a freely-available software package. It is available from netlib via anonymous ftp and the World Wide Web.

•

LAPACK. The complete LAPACK package or individual routines from LAPACK are freely available on netlib and can be obtained via the World Wide Web or anonymous ftp.

•

For maximal benefit to the community, we added the sources we are proprietary of to the tar.gz file submitted for inclusion in the CPC library. However, as explained in the README file, users willing to compile the code instead of using binaries should first obtain the sources for the external packages mentioned above (email and/or web addresses are provided).

Running time: Problem dependent; the test examples provided with the code only take a few seconds to run; timing results for large scale problems are given in Section 5.     

Computation of the eigenvalues of the Schrödinger equation by exponentially-fitted Runge-Kutta-Nyström methods

In this work we consider exponentially fitted and trigonometrically fitted Runge-Kutta-Nyström methods. These methods integrate exactly differential systems whose solutions can be expressed as linear combinations of the set of functions exp(wx), exp(−wx), or sin(wx), cos(wx), w∈ℜ. We modify existing RKN methods of fifth and sixth order. We apply these methods to the computation of the eigenvalues of the Schrödinger equation with different potentials as the harmonic oscillator, the doubly anharmonic oscillator and the exponential potential.     

Gauss-Legendre and Chebyshev quadratures for singular integrals

Exact expressions are presented for efficient computation of the weights in Gauss-Legendre and Chebyshev quadratures for selected singular integrands. The singularities may be of Cauchy type, logarithmic type or algebraic-logarithmic end-point branching points. We provide Fortran 90 routines for computing the weights for both the Gauss-Legendre and the Chebyshev (Fejér-1) meshes whose size can be set by the user.

New program summary

Program title: SINGQUADCatalogue identifier: AEBR_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEBR_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.: 4128No. of bytes in distributed program, including test data, etc.: 25 815Distribution format: tar.gzProgramming language: Fortran 90Computer: Any with a Fortran 90 compilerOperating system: Linux, Windows, MacRAM: Depending on the complexity of the problemClassification: 4.11Nature of problem: Program provides Gauss-Legendre and Chebyshev (Fejér-1) weights for various singular integrands.Solution method: The weights are obtained from the condition that the quadrature of order N must be exact for a polynomial of degree?(N−1). The weights are expressed as moments of the singular kernels associated with Legendre or Chebyshev polynomials. These moments are obtained in analytic form amenable for computation.Additional comments: If the NAGWare f95 compiler is used, the option, “-kind = byte”, must be included in the compile command lines of the Makefile.Running time: The test run supplied with the distribution takes a couple of seconds to execute.     

POTHMF: A program for computing potential curves and matrix elements of the coupled adiabatic radial equations for a hydrogen-like atom in a homogeneous magnetic field

A FORTRAN 77 program is presented which calculates with the relative machine precision potential curves and matrix elements of the coupled adiabatic radial equations for a hydrogen-like atom in a homogeneous magnetic field. The potential curves are eigenvalues corresponding to the angular oblate spheroidal functions that compose adiabatic basis which depends on the radial variable as a parameter. The matrix elements of radial coupling are integrals in angular variables of the following two types: product of angular functions and the first derivative of angular functions in parameter, and product of the first derivatives of angular functions in parameter, respectively. The program calculates also the angular part of the dipole transition matrix elements (in the length form) expressed as integrals in angular variables involving product of a dipole operator and angular functions. Moreover, the program calculates asymptotic regular and irregular matrix solutions of the coupled adiabatic radial equations at the end of interval in radial variable needed for solving a multi-channel scattering problem by the generalized R-matrix method. Potential curves and radial matrix elements computed by the POTHMF program can be used for solving the bound state and multi-channel scattering problems. As a test desk, the program is applied to the calculation of the energy values, a short-range reaction matrix and corresponding wave functions with the help of the KANTBP program. Benchmark calculations for the known photoionization cross-sections are presented.

Program summary

Program title:POTHMFCatalogue identifier:AEAA_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEAA_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.:8123No. of bytes in distributed program, including test data, etc.:131 396Distribution format:tar.gzProgramming language:FORTRAN 77Computer:Intel Xeon EM64T, Alpha 21264A, AMD Athlon MP, Pentium IV Xeon, Opteron 248, Intel Pentium IVOperating system:OC Linux, Unix AIX 5.3, SunOS 5.8, Solaris, Windows XPRAM:Depends on

1.

the number of radial differential equations;

2.

the number and order of finite elements;

3.

the number of radial points.

Test run requires 4 MBClassification:2.5External routines:POTHMF uses some Lapack routines, copies of which are included in the distribution (see README file for details).Nature of problem:In the multi-channel adiabatic approach the Schrödinger equation for a hydrogen-like atom in a homogeneous magnetic field of strength γ (γ=B/B₀, B₀≅2.35×¹⁰⁵ T is a dimensionless parameter which determines the field strength B) is reduced by separating the radial coordinate, r, from the angular variables, (θ,φ), and using a basis of the angular oblate spheroidal functions [3] to a system of second-order ordinary differential equations which contain first-derivative coupling terms [4]. The purpose of this program is to calculate potential curves and matrix elements of radial coupling needed for calculating the low-lying bound and scattering states of hydrogen-like atoms in a homogeneous magnetic field of strength 0<γ?1000 within the adiabatic approach [5]. The program evaluates also asymptotic regular and irregular matrix radial solutions of the multi-channel scattering problem needed to extract from the R-matrix a required symmetric shortrange open-channel reaction matrix K [6] independent from matching point [7]. In addition, the program computes the dipole transition matrix elements in the length form between the basis functions that are needed for calculating the dipole transitions between the low-lying bound and scattering states and photoionization cross sections [8].Solution method:The angular oblate spheroidal eigenvalue problem depending on the radial variable is solved using a series expansion in the Legendre polynomials [3]. The resulting tridiagonal symmetric algebraic eigenvalue problem for the evaluation of selected eigenvalues, i.e. the potential curves, is solved by the LDL^T factorization using the DSTEVR program [2]. Derivatives of the eigenfunctions with respect to the radial variable which are contained in matrix elements of the coupled radial equations are obtained by solving the inhomogeneous algebraic equations. The corresponding algebraic problem is solved by using the LDL^T factorization with the help of the DPTTRS program [2]. Asymptotics of the matrix elements at large values of radial variable are computed using a series expansion in the associated Laguerre polynomials [9]. The corresponding matching points between the numeric and asymptotic solutions are found automatically. These asymptotics are used for the evaluation of the asymptotic regular and irregular matrix radial solutions of the multi-channel scattering problem [7]. As a test desk, the program is applied to the calculation of the energy values of the ground and excited bound states and reaction matrix of multi-channel scattering problem for a hydrogen atom in a homogeneous magnetic field using the KANTBP program [10].Restrictions:The computer memory requirements depend on:

1.

the number of radial differential equations;

2.

the number and order of finite elements;

3.

the total number of radial points.

Restrictions due to dimension sizes can be changed by resetting a small number of PARAMETER statements before recompiling (see Introduction and listing for details).Running time:The running time depends critically upon:

1.

the number of radial differential equations;

2.

the number and order of finite elements;

3.

the total number of radial points on interval [rmin,rmax].

The test run which accompanies this paper took 7 s required for calculating of potential curves, radial matrix elements, and dipole transition matrix elements on a finite-element grid on interval [rmin=0, rmax=100] used for solving discrete and continuous spectrum problems and obtaining asymptotic regular and irregular matrix radial solutions at rmax=100 for continuous spectrum problem on the Intel Pentium IV 2.4 GHz. The number of radial differential equations was equal to 6. The accompanying test run using the KANTBP program took 2 s for solving discrete and continuous spectrum problems using the above calculated potential curves, matrix elements and asymptotic regular and irregular matrix radial solutions. Note, that in the accompanied benchmark calculations of the photoionization cross-sections from the bound states of a hydrogen atom in a homogeneous magnetic field to continuum we have used interval [rmin=0, rmax=1000] for continuous spectrum problem. The total number of radial differential equations was varied from 10 to 18.References:

[1]

W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, Cambridge, 1986.

[2]

http://www.netlib.org/lapack/.

[3]

M. Abramovits, I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965.

[4]

U. Fano, Colloq. Int. C.N.R.S. 273 (1977) 127; A.F. Starace, G.L. Webster, Phys. Rev. A 19 (1979) 1629-1640; C.V. Clark, K.T. Lu, A.F. Starace, in: H.G. Beyer, H. Kleinpoppen (Eds.), Progress in Atomic Spectroscopy, Part C, Plenum, New York, 1984, pp. 247-320; U. Fano, A.R.P. Rau, Atomic Collisions and Spectra, Academic Press, Florida, 1986.

[5]

M.G. Dimova, M.S. Kaschiev, S.I. Vinitsky, J. Phys. B 38 (2005) 2337-2352; O. Chuluunbaatar, A.A. Gusev, V.L. Derbov, M.S. Kaschiev, V.V. Serov, T.V. Tupikova, S.I. Vinitsky, Proc. SPIE 6537 (2007) 653706-1-18.

[6]

M.J. Seaton, Rep. Prog. Phys. 46 (1983) 167-257.

[7]

M. Gailitis, J. Phys. B 9 (1976) 843-854; J. Macek, Phys. Rev. A 30 (1984) 1277-1278; S.I. Vinitsky, V.P. Gerdt, A.A. Gusev, M.S. Kaschiev, V.A. Rostovtsev, V.N. Samoylov, T.V. Tupikova, O. Chuluunbaatar, Programming and Computer Software 33 (2007) 105-116.

[8]

H. Friedrich, Theoretical Atomic Physics, Springer, New York, 1991.

[9]

R.J. Damburg, R.Kh. Propin, J. Phys. B 1 (1968) 681-691; J.D. Power, Phil. Trans. Roy. Soc. London A 274 (1973) 663-702.

[10]

O. Chuluunbaatar, A.A. Gusev, A.G. Abrashkevich, A. Amaya-Tapia, M.S. Kaschiev, S.Y. Larsen, S.I. Vinitsky, Comput. Phys. Comm. 177 (2007) 649-675.

     

CANM, a program for numerical solution of a system of nonlinear equations using the continuous analog of Newton's method

A FORTRAN program is presented which solves a system of nonlinear simultaneous equations using the continuous analog of Newton's method (CANM). The user has the option of either to provide a subroutine which calculates the Jacobian matrix or allow the program to calculate it by a forward-difference approximation. Five iterative schemes using different algorithms of determining adaptive step size of the CANM process are implemented in the program.

Program summary

Title of program: CANMCatalogue number: ADSNProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADSNProgram available from: CPC Program Library, Queen's University of Belfast, Northern IrelandLicensing provisions: noneComputer for which the program is designed and others on which it has been tested:Computers: IBM RS/6000 Model 320H, SGI Origin2000, SGI Octane, HP 9000/755, Intel Pentium IV PCInstallation: Department of Chemistry, University of Toronto, Toronto, CanadaOperating systems under which the program has been tested: IRIX64 6.1, 6.4 and 6.5, AIX 3.4, HP-UX 9.01, Linux 2.4.7Programming language used: FORTRAN 90Memory required to execute with typical data: depends on the number of nonlinear equations in a system. Test run requires 80 KBNo. of bits in distributed program including test data, etc.: 15283Distribution format: tar gz formatNo. of lines in distributed program, including test data, etc.: 1794Peripherals used: line printer, scratch disc storeExternal subprograms used: DGECO and DGESL [1]Keywords: nonlinear equations, Newton's method, continuous analog of Newton's method, continuous parameter, evolutionary differential equation, Euler's methodNature of physical problem: System of nonlinear simultaneous equations

     

Efficient computation of high index Sturm-Liouville eigenvalues for problems in physics

Finding the eigenvalues of a Sturm-Liouville problem can be a computationally challenging task, especially when a large set of eigenvalues is computed, or just when particularly large eigenvalues are sought. This is a consequence of the highly oscillatory behavior of the solutions corresponding to high eigenvalues, which forces a naive integrator to take increasingly smaller steps. We will discuss the most used approaches to the numerical solution of the Sturm-Liouville problem: finite differences and variational methods, both leading to a matrix eigenvalue problem; shooting methods using an initial-value solver; and coefficient approximation methods. Special attention will be paid to techniques that yield uniform approximation over the whole eigenvalue spectrum and that allow large steps even for high eigenvalues.     

NumSBT: A subroutine for calculating spherical Bessel transforms numerically

A previous subroutine, LSFBTR, for computing numerical spherical Bessel (Hankel) transforms is updated with several improvements and modifications. The procedure is applicable if the input radial function and the output transform are defined on logarithmic meshes and if the input function satisfies reasonable smoothness conditions. Important aspects of the procedure are that it is simply implemented with two successive applications of the fast Fourier transform, and it yields accurate results at very large values of the transform variable. Applications to the evaluation of overlap integrals and the Coulomb potential of multipolar charge distributions are described.

Program summary

Program title: NumSBTCatalogue identifier: AANZ_v2_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AANZ_v2_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.: 476No. of bytes in distributed program, including test data, etc.: 4451Distribution format: tar.gzProgramming language: Fortran 90Computer: GenericOperating system: LinuxClassification: 4.6Catalogue identifier of previous version: AANZ_v1_0Journal reference of previous version: Comput. Phys. Comm. 30 (1983) 93Does the new version supersede the previous version?: NoNature of problem: This program is a subroutine which, for a function defined numerically on a logarithmic mesh in the radial coordinate, generates the spherical Bessel, or Hankel, transform on a logarithmic mesh in the transform variable. Accurate results for large values of the transform variable are obtained, that would not be otherwise obtainable.Solution method: The program applies a procedure proposed by the author [1] that treats the problem as a convolution. The calculation then requires two applications of the fast Fourier transform method.Reasons for new version: The method of computing the transform at small values of the transform variable has been substantially changed and the whole procedure simplified. In addition, the possibility of computing the transform for a single transform variable value has been incorporated. The code has also been converted to Fortran 90 from Fortran 77.Restrictions: The procedure is most applicable to smooth functions defined on (0,∞) with a limited number of nodes.Running time: The example provided with the distribution takes a few seconds to execute.References:[1] J.D. Talman, J. Comp. Phys. 29 (1978) 35.     

On the evaluation of H and X functions

We indicate a Double Exponential Formula based numerical integration method for the evaluation of the Ambarzumian-Chandrasekhar H function and the X function of neutron transport for the single speed and isotropic case. This method is significantly more economical than our earlier scheme, which was based on IMT quadrature. For c<5, the present method converges faster than our earlier IMT scheme. This will be adequate for all radiative transport and transport theory applications. These findings are supported by appropriate error analysis. Unlike the IMT method, the DE quadrature nodes are generated by a simple algebraic expression which is a great advantage.     

PANMIN: sequential and parallel global optimization procedures with a variety of options for the local search strategy

We present two sequential and one parallel global optimization codes, that belong to the stochastic class, and an interface routine that enables the use of the Merlin/MCL environment as a non-interactive local optimizer. This interface proved extremely important, since it provides flexibility, effectiveness and robustness to the local search task that is in turn employed by the global procedures. We demonstrate the use of the parallel code to a molecular conformation problem.

Program summary

Title of program: PANMINCatalogue identifier: ADSUProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADSUProgram 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: PANMIN is designed for UNIX machines. The parallel code runs on either shared memory architectures or on a distributed system. The code has been tested on a SUN Microsystems ENTERPRISE 450 with four CPUs, and on a 48-node cluster under Linux, with both the GNU g77 and the Portland group compilers. The parallel implementation is based on MPI and has been tested with LAM MPI and MPICHInstallation: University of Ioannina, GreeceProgramming language used: Fortran-77Memory required to execute with typical data: Approximately O(n²) words, where n is the number of variablesNo. of bits in a word: 64No. of processors used: 1 or manyHas the code been vectorised or parallelized?: Parallelized using MPINo. of bytes in distributed program, including test data, etc.: 147163No. of lines in distributed program, including the test data, etc.: 14366Distribution format: gzipped tar fileNature of physical problem: A multitude of problems in science and engineering are often reduced to minimizing a function of many variables. There are instances that a local optimum does not correspond to the desired physical solution and hence the search for a better solution is required. Local optimization techniques can be trapped in any local minimum. Global Optimization is then the appropriate tool. For example, solving a non-linear system of equations via optimization, one may encounter many local minima that do not correspond to solutions, i.e. they are far from zeroMethod of solution: PANMIN is a suite of programs for Global Optimization that take advantage of the Merlin/MCL optimization environment [1,2]. We offer implementations of two algorithms that belong to the stochastic class and use local searches either as intermediate steps or as solution refinementRestrictions on the complexity of the problem: The only restriction is set by the available memory of the hardware configuration. The software can handle bound constrained problems. The Merlin Optimization environment must be installed. Availability of an MPI installation is necessary for executing the parallel codeTypical running time: Depending on the objective functionReferences: [1] D.G. Papageorgiou, I.N. Demetropoulos, I.E. Lagaris, Merlin-3.0. A multidimensional optimization environment, Comput. Phys. Commun. 109 (1998) 227-249. [2] D.G. Papageorgiou, I.N. Demetropoulos, I.E. Lagaris, The Merlin Control Language for strategic optimization, Comput. Phys. Commun. 109 (1998) 250-275.