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

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

GYutsis: heuristic based calculation of general recoupling coefficients

General angular momentum recoupling coefficients can be expressed as a summation formula over products of 6-j coefficients. Yutsis, Levinson and Vanagas developed graphical techniques for representing the general recoupling coefficient as a cubic graph and they describe a set of reduction rules allowing a stepwise generation of the corresponding summation formula. This paper is a follow up to [Van Dyck and Fack, Comput. Phys. Comm. 151 (2003) 353-368] where we described a heuristic algorithm based on these techniques. In this article we separate the heuristic from the algorithm and describe some new heuristic approaches which can be plugged into the generic algorithm. We show that these new heuristics lead to good results: in many cases we get a more efficient summation formula than our previous approach, in particular for problems of higher order. In addition the new features and the use of our program GYutsis, which implements these techniques, is described both for end users and application programmers.

Program summary

Title of program: CycleCostAlgorithm, GYutsisCatalogue number: ADSAProgram Summary URL:http://cpc.cs.qub.ac.uk/summaries/ADSAProgram obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland. Users may obtain the program also by downloading either the compressed tar file gyutsis.tgz (for Unix and Linux) or the zip file gyutsis.zip (for Windows) from our website (http://caagt.rug.ac.be/yutsis/). An applet version of the program is also available on our website and can be run in a web browser from the URL http://caagt.rug.ac.be/yutsis/GYutsisApplet.html.Licensing provisions: noneComputers for which the program is designed: any computer with Sun's Java Runtime Environment 1.4 or higher installed.Programming language used: Java 1.2 (Compiler: Sun's SDK 1.4.0)No. of lines in program: approximately 9400No. of bytes in distributed program, including test data, etc.: 544 117Distribution format: tar gzip fileNature of physical problem: A general recoupling coefficient for an arbitrary number of (integer or half-integer) angular momenta can be expressed as a formula consisting of products of 6-j coefficients summed over a certain number of variables. Such a formula can be generated using the program GYutsis (with a graphical user front end) or CycleCostAlgorithm (with a text-mode user front end).Method of solution: Using the graphical techniques of Yutsis, Levinson and Vanagas (1962) a summation formula for a general recoupling coefficient is obtained by representing the coefficient as a Yutsis graph and by performing a selection of reduction rules valid for such graphs. Each reduction rule contributes to the final summation formula by a numerical factor or by an additional summation variable. Whereas an optimal summation formula (i.e. with a minimum number of summation variables) is hard to obtain, we present here some new heuristic approaches for selecting an edge from a k-cycle in order to transform it into an (k−1)-cycle (k>3) in such a way that a ‘good’ summation formula is obtained.Typical running time: From instantaneously for the typical problems to 30 s for the heaviest problems on a Pentium II-350 Linux-system with 256 MB RAM.     

Optimized implementations of rational approximations—a case study on the Voigt and complex error function

Rational functions are frequently used as efficient yet accurate numerical approximations for real and complex valued special functions. For the complex error function w(x+iy), whose real part is the Voigt function K(x,y), the rational approximation developed by Hui, Armstrong, and Wray [Rapid computation of the Voigt and complex error functions, J. Quant. Spectrosc. Radiat. Transfer 19 (1978) 509-516] is investigated. Various optimizations for the algorithm are discussed. In many applications, where these functions have to be calculated for a large x grid with constant y, an implementation using real arithmetic and factorization of invariant terms is especially efficient.     

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

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

Program summary

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

A Mathematica program for the two-step twelfth-order method with multi-derivative for the numerical solution of a one-dimensional Schrödinger equation

In this paper, we present the detailed Mathematica symbolic derivation and the program which is used to integrate a one-dimensional Schrödinger equation by a new two-step numerical method. We add the fourth- and sixth-order derivatives to raise the precision of the traditional Numerov's method from fourth order to twelfth order, and to expand the interval of periodicity from (0,6) to the one of (0,9.7954) and (9.94792,55.6062). In the program we use an efficient algorithm to calculate the first-order derivative and avoid unnecessarily repeated calculation resulting from the multi-derivatives. We use the well-known Woods-Saxon's potential to test our method. The numerical test shows that the new method is not only superior to the previous lower order ones in accuracy, but also in the efficiency. This program is specially applied to the problem where a high accuracy or a larger step size is required.

Program summary

Title of program: ShdEq.nbCatalogue number: ADTTProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADTTProgram obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandLicensing provisions: noneComputer for which the program is designed and others on which it has been tested: The program has been designed for the microcomputer and been tested on the microcomputer.Computers: IBM PCOperating systems under which the program has been tested: Windows XPProgramming language used: Mathematica 4.2Memory required to execute with typical data: 51 712 bytesNo. of bytes in distributed program, including test data, etc.: 45 381No. of lines in distributed program, including test data, etc.: 7311Distribution format: tar gzip fileCPC Program Library subprograms used: noNature of physical problem: Numerical integration of one-dimensional or radial Schrödinger equation to find the eigenvalues for a bound states and phase shift for a continuum state.Method of solution: Using a two-step method twelfth-order method to integrate a Schrödinger equation numerically from both two ends and the connecting conditions at the matching point, an eigenvalue for a bound state or a resonant state with a given phase shift can be found.Restrictions on the complexity of the problem: The analytic form of the potential function and its high-order derivatives must be known.Typical running time: Less than one second.Unusual features of the program: Take advantage of the high-order derivatives of the potential function and efficient algorithm, the program can provide all the numerical solution of a given Schrödinger equation, either a bound or a resonant state, with a very high precision and within a very short CPU time. The program can apply to a very broad range of problems because the method has a very large interval of periodicity.References: [1] T.E. Simos, Proc. Roy. Soc. London A 441 (1993) 283.[2] Z. Wang, Y. Dai, An eighth-order two-step formula for the numerical integration of the one-dimensional Schrödinger equation, Numer. Math. J. Chinese Univ. 12 (2003) 146.[3] Z. Wang, Y. Dai, An twelfth-order four-step formula for the numerical integration of the one-dimensional Schrödinger equation, Internat. J. Modern Phys. C 14 (2003) 1087.     

Qprop: A Schrödinger-solver for intense laser-atom interaction

The Qprop package is presented. Qprop has been developed to study laser-atom interaction in the nonperturbative regime where nonlinear phenomena such as above-threshold ionization, high order harmonic generation, and dynamic stabilization are known to occur. In the nonrelativistic regime and within the single active electron approximation, these phenomena can be studied with Qprop in the most rigorous way by solving the time-dependent Schrödinger equation in three spatial dimensions. Because Qprop is optimized for the study of quantum systems that are spherically symmetric in their initial, unperturbed configuration, all wavefunctions are expanded in spherical harmonics. Time-propagation of the wavefunctions is performed using a split-operator approach. Photoelectron spectra are calculated employing a window-operator technique. Besides the solution of the time-dependent Schrödinger equation in single active electron approximation, Qprop allows to study many-electron systems via the solution of the time-dependent Kohn-Sham equations.

Program summary

Program title:QPROPCatalogue number:ADXBProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADXBProgram obtainable from:CPC Program Library, Queen's University of Belfast, N. IrelandComputer on which program has been tested:PC Pentium IV, AthlonOperating system:LinuxProgram language used:C++Memory required to execute with typical data:Memory requirements depend on the number of propagated orbitals and on the size of the orbitals. For instance, time-propagation of a hydrogenic wavefunction in the perturbative regime requires about 64 KB RAM (4 radial orbitals with 1000 grid points). Propagation in the strongly nonperturbative regime providing energy spectra up to high energies may need 60 radial orbitals, each with 30000 grid points, i.e. about 30 MB. Examples are given in the article.No. of bits in a word:Real and complex valued numbers of double precision are usedNo. of lines in distributed program, including test data, etc.:69 995No. of bytes in distributed program, including test data, etc.: 2 927 567Peripheral used:Disk for input-output, terminal for interaction with the userCPU time required to execute test data:Execution time depends on the size of the propagated orbitals and the number of time-stepsDistribution format:tar.gzNature of the physical problem:Atoms put into the strong field of modern lasers display a wealth of novel phenomena that are not accessible to conventional perturbation theory where the external field is considered small as compared to inneratomic forces. Hence, the full ab initio solution of the time-dependent Schrödinger equation is desirable but in full dimensionality only feasible for no more than two (active) electrons. If many-electron effects come into play or effective ground state potentials are needed, (time-dependent) density functional theory may be employed. Qprop aims at providing tools for (i) the time-propagation of the wavefunction according to the time-dependent Schrödinger equation, (ii) the time-propagation of Kohn-Sham orbitals according to the time-dependent Kohn-Sham equations, and (iii) the energy-analysis of the final one-electron wavefunction (or the Kohn-Sham orbitals).Method of solution:An expansion of the wavefunction in spherical harmonics leads to a coupled set of equations for the radial wavefunctions. These radial wavefunctions are propagated using a split-operator technique and the Crank-Nicolson approximation for the short-time propagator. The initial ground state is obtained via imaginary time-propagation for spherically symmetric (but otherwise arbitrary) effective potentials. Excited states can be obtained through the combination of imaginary time-propagation and orthogonalization. For the Kohn-Sham scheme a multipole expansion of the effective potential is employed. Wavefunctions can be analyzed using the window-operator technique, facilitating the calculation of electron spectra, either angular-resolved or integratedRestrictions onto the complexity of the problem:The coupling of the atom to the external field is treated in dipole approximation. The time-dependent Schrödinger solver is restricted to the treatment of a single active electron. As concerns the time-dependent density functional mode of Qprop, the Hartree-potential (accounting for the classical electron-electron repulsion) is expanded up to the quadrupole. Only the monopole term of the Krieger-Li-Iafrate exchange potential is currently implemented. As in any nontrivial optimization problem, convergence to the optimal many-electron state (i.e. the ground state) is not automatically guaranteedExternal routines/libraries used:The program uses the well established libraries blas, lapack, and f2c     

An object-oriented C++ implementation of Davidson method for finding a few selected extreme eigenpairs of a large, sparse, real, symmetric matrix

A C++ class named Davidson is presented for determining a few eigenpairs with lowest or alternatively highest values of a large, real, symmetric matrix. The algorithm described by Stathopoulos and Fischer is used. The exception mechanism is involved to report the errors. The class is written in ANSI C++, so it is fully portable. In addition a console program as well as a program with graphical user interface for Microsoft Windows is attached, which allow one to calculate the lowest eigenstates of time-independent Schrödinger equation for a given binding potential in one, two or three spatial dimensions. The package contains the classes providing often used potential functions (model atom potential, Coulomb potential, square well potential and Kramers-Henneberger well potential) as well as a possibility to use any potential stored in a file (then any dimensionality of the problem is allowed).The described code is the subject of M.Sc. thesis of T.D. prepared under the supervision of J.M.

Program summary

Program title: DavidsonCatalogue identifier: ADZM_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADZM_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.: 3 037 055No. of bytes in distributed program, including test data, etc.: 20 002 609Distribution format: tar.gzProgramming language: C++Computer: AllOperating system: AnyRAM: User's parameters dependentWord size: 32 and 64 bitsSupplementary material: Test results for the 2D and 3D cases is availableClassification: 4, 4.8Nature of problem: Finding a few extreme eigenpairs of a real, symmetric, sparse matrix. Examples in quantum optics (interaction of matter with a laser field).Solution method: Davidson algorithmRunning time: The test example included in the distribution package (1D matrix) takes approximately 30 minutes to run. 2D matrix calculations can take hours and 3D, days, to run.     

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.     

FFT-split-operator code for solving the Dirac equation in 2+1 dimensions

The main part of the code presented in this work represents an implementation of the split-operator method [J.A. Fleck, J.R. Morris, M.D. Feit, Appl. Phys. 10 (1976) 129-160; R. Heather, Comput. Phys. Comm. 63 (1991) 446] for calculating the time-evolution of Dirac wave functions. It allows to study the dynamics of electronic Dirac wave packets under the influence of any number of laser pulses and its interaction with any number of charged ion potentials. The initial wave function can be either a free Gaussian wave packet or an arbitrary discretized spinor function that is loaded from a file provided by the user. The latter option includes Dirac bound state wave functions. The code itself contains the necessary tools for constructing such wave functions for a single-electron ion. With the help of self-adaptive numerical grids, we are able to study the electron dynamics for various problems in 2+1 dimensions at high spatial and temporal resolutions that are otherwise unachievable.Along with the position and momentum space probability density distributions, various physical observables, such as the expectation values of position and momentum, can be recorded in a time-dependent way. The electromagnetic spectrum that is emitted by the evolving particle can also be calculated with this code. Finally, for planning and comparison purposes, both the time-evolution and the emission spectrum can also be treated in an entirely classical relativistic way.Besides the implementation of the above-mentioned algorithms, the program also contains a large C++ class library to model the geometric algebra representation of spinors that we use for representing the Dirac wave function. This is why the code is called “Dirac++”.

Program summary

Program title: Dirac++ or (abbreviated) d++Catalogue identifier: AEAS_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEAS_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.: 474 937No. of bytes in distributed program, including test data, etc.: 4 128 347Distribution format: tar.gzProgramming language: C++Computer: Any, but SMP systems are preferredOperating system: Linux and MacOS X are actively supported by the current version. Earlier versions were also tested successfully on IRIX and AIXNumber of processors used: Generally unlimited, but best scaling with 2-4 processors for typical problemsRAM: 160 Megabytes minimum for the examples given hereClassification: 2.7External routines: FFTW Library [3,4], Gnu Scientific Library [5], bzip2, bunzip2Nature of problem: The relativistic time evolution of wave functions according to the Dirac equation is a challenging numerical task. Especially for an electron in the presence of high intensity laser beams and/or highly charged ions, this type of problem is of considerable interest to atomic physicists.Solution method: The code employs the split-operator method [1,2], combined with fast Fourier transforms (FFT) for calculating any occurring spatial derivatives, to solve the given problem. An autocorrelation spectral method [6] is provided to generate a bound state for use as the initial wave function of further dynamical studies.Restrictions: The code in its current form is restricted to problems in two spatial dimensions. Otherwise it is only limited by CPU time and memory that one can afford to spend on a particular problem.Unusual features: The code features dynamically adapting position and momentum space grids to keep execution time and memory requirements as small as possible. It employs an object-oriented approach, and it relies on a Clifford algebra class library to represent the mathematical objects of the Dirac formalism which we employ. Besides that it includes a feature (typically called “checkpointing”) which allows the resumption of an interrupted calculation.Additional comments: Along with the program's source code, we provide several sample configuration files, a pre-calculated bound state wave function, and template files for the analysis of the results with both MatLab and Igor Pro.Running time: Running time ranges from a few minutes for simple tests up to several days, even weeks for real-world physical problems that require very large grids or very small time steps.References:

[1]

J.A. Fleck, J.R. Morris, M.D. Feit, Time-dependent propagation of high energy laser beams through the atmosphere, Appl. Phys. 10 (1976) 129-160.

[2]

R. Heather, An asymptotic wavefunction splitting procedure for propagating spatially extended wavefunctions: Application to intense field photodissociation of H⁺₂, Comput. Phys. Comm. 63 (1991) 446.

[3]

M. Frigo, S.G. Johnson, FFTW: An adaptive software architecture for the FFT, in: Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing, vol. 3, IEEE, 1998, pp. 1381-1384.

[4]

M. Frigo, S.G. Johnson, The design and implementation of FFTW3, in: Proceedings of the IEEE, vol. 93, IEEE, 2005, pp. 216-231. URL: http://www.fftw.org/.

[5]

M. Galassi, J. Davies, J. Theiler, B. Gough, G. Jungman, M. Booth, F. Rossi, GNU Scientific Library Reference Manual, second ed., Network Theory Limited, 2006. URL: http://www.gnu.org/software/gsl/.

[6]

M.D. Feit, J.A. Fleck, A. Steiger, Solution of the Schrödinger equation by a spectral method, J. Comput. Phys. 47 (1982) 412-433.

     

Trigonometrically-fitted method with the Fourier frequency spectrum for undamped Duffing equation

With non-linearities, the frequency spectrum of an undamped Duffing oscillator should be composed of odd multiples of the driving frequency which can be interpreted as resonance driving terms. It is expected that the frequency spectrum of the corresponding numerical solution with high accurateness should contain nearly the same components. Hence, to contain these Fourier components and to calculate the amplitudes of these components in a more accurate and efficient way is the key to develop a new numerical method with high stability, accuracy and efficiency for the Duffing equation. To explore the possibility of using trigonometrically-fitting technique to build a numerical method with resonance spectrum, we design four types of Numerov methods, in which the first one is the traditional Numerov method, which contains no Fourier component, the second one contains only the first resonance term, the third one contains the first two resonance terms, and the last one contains the first three resonance terms, and apply them to the well-known undamped Duffing equation with Dooren's parameters. The numerical results demonstrate that the Numerov method fitted with the Fourier components is much more stable, accurate and efficient than the one with no Fourier component. The accuracy of the fitted method with the first three Fourier components can attain 10⁻⁹ for a remarkable range of step sizes, including nearly infinite, except individual small range of instability, which is much higher than the one of the traditional Numerov method, with eight orders for step size of π/2.011.     

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^′.     

New heuristic approach to the calculation of general recoupling coefficients

General angular momentum recoupling coefficients can be expressed as a summation formula over products of 6-j coefficients. Yutsis, Levinson and Vanagas developed graphical techniques for representing the general recoupling coefficient as a cubic graph and they describe a set of reduction rules on such a graph which allow a stepwise generation of the corresponding summation formula. In this paper we present a new heuristic approach to earlier reduction algorithms based on these techniques. In particular the heuristic tries in each step to make an ‘intelligent’ choice of which specific reduction rule to apply on which part of the graph. This approach leads to good results: in many cases we get a more efficient summation formula than the other algorithms. The program is written in Java and offers a platform-independent graphical user front end which allows user-friendly manipulation of recoupling coefficients as well as a detailed follow-up of the reduction process.     

Fast spherical Bessel transform via fast Fourier transform and recurrence formula

We propose a new method for the numerical evaluation of the spherical Bessel transform. A formula is derived for the transform by using an integral representation of the spherical Bessel function and by changing the integration variable. The resultant algorithm consists of a set of the Fourier transforms and numerical integrations over a linearly spaced grid of variable k in Fourier space. Because the k-dependence appears in the upper limit of the integration range, the integrations can be performed effectively in a recurrence formula. Several types of atomic orbital functions are transformed with the proposed method to illustrate its accuracy and efficiency, demonstrating its applicability for transforms of general order with high accuracy.     

A Mathematica program for the approximate analytical solution to a nonlinear undamped Duffing equation by a new approximate approach

According to Mickens [R.E. Mickens, Comments on a Generalized Galerkin's method for non-linear oscillators, J. Sound Vib. 118 (1987) 563], the general HB (harmonic balance) method is an approximation to the convergent Fourier series representation of the periodic solution of a nonlinear oscillator and not an approximation to an expansion in terms of a small parameter. Consequently, for a nonlinear undamped Duffing equation with a driving force Bcos(ωx), to find a periodic solution when the fundamental frequency is identical to ω, the corresponding Fourier series can be written as

     

A numerical method for diffusion-convection equation using high-order difference schemes

In this paper, we propose a simple general form of high-order approximation of O(c²+ch²+h⁴) to solve the two-dimensional parabolic equation αuxx+βuyy=F(x,y,t,u,ux,uy,ut), where α and β are positive constants. We apply the compact form for solving diffusion-convection equation. The results of numerical experiments are presented and compared with analytical solutions to confirm the higher accuracy of the presented scheme.     

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.     

Numerical errors of explicit finite difference approximation for two-dimensional solute transport equation with linear sorption

The numerical errors associated with explicit upstream finite difference solutions of two-dimensional advection—Dispersion equation with linear sorption are formulated from a Taylor analysis. The error expressions are based on a general form of the corresponding difference equation. The numerical truncation errors are defined using Peclet and Courant numbers in the X and Y direction, a sink/source dimensionless number and new Peclet and Courant numbers in the XY plane. The effects of these truncation errors on the explicit solution of a two-dimensional advection–dispersion equation with a first-order reaction or degradation are demonstrated by comparison with an analytical solution in uniform flow field. The results show that these errors are not negligible and correcting the finite difference scheme for them results in a more accurate solution.