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  • 1.
    Edvardsson, Sverker
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences.
    Karlsson, Kristoffer
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Science Education and Mathematics.
    Olin, Håkan
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences.
    corr3p_tr: A particle approach for the general three-body problem2016In: Computer Physics Communications, ISSN 0010-4655, E-ISSN 1879-2944, Vol. 200, p. 259-273Article in journal (Refereed)
    Abstract [en]

    This work presents a convenient way to solve the non-relativistic Schrodinger equation numerically for a general three-particle system including full correlation and mass polarization. Both Coulombic and non-Coulombic interactions can be studied. The eigensolver is based on a second order dynamical system treatment (particle method). The Hamiltonian matrix never needs to be realized. The wavefunction evolves towards the steady state solution for which the Schrodinger equation is fulfilled. Subsequent Richardson extrapolations for several meshes are then made symbolically in matlab to obtain the continuum solution. The computer C code is tested under Linux 64 bit and both double and extended precision versions are provided. Test runs are exemplified and, when possible, compared with corresponding values in the literature. The computer code is small and self contained making it unusually simple to compile and run on any system. Both serial and parallel computer runs are straight forward. Program summary Program title: corr3p_tr Catalogue identifier: AEYR_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEYR_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.ukilicence/licence.html No. of lines in distributed program, including test data, etc.: 15025 No. of bytes in distributed program, including test data, etc.: 156430 Distribution format: tar.gz Programming language: ANSI C. Computer: Linux 64bit PC. Operating system: Linux 64bit. RAM: 300 M bytes Classification: 2.7, 2.8, 2.9. Nature of problem: The Schrodinger equation for an arbitrary three -particle system is solved using finite differences and a fast particle method for the eigenvalue problem [20, 21, 23]. Solution method: A fast eigensolver is applied (see Appendix). This solver works for both symmetrical and nonsymmetrical matrices (which opens up for more accurate nonsymmetrical finite difference expressions to be applied at the boundaries). The three-particle Schrodinger equation is transformed in two major steps. First step is to introduce the function Q(r(1), (r)2, mu) = r(1)r(2)(1 - mu(2))phi(r(1), r(2), mu), where mu = cos (0(12)). The cusps (r(1) = r(2), mu = 1) are then transformed into boundary conditions. The derivatives of Qare then continuous in the whole computational space and thus the finite difference expressions are well defined. Three-particle coalescence (r(1) = r(2) = 0, mu) is treated in the same way. The second step is to replace Q(r(1), r(2), mu) with (2,root x(1)x(2))(-1)Q(x(1) x(2), mu). The space (x(1), x(2), mu) is much more appropriate for a finite difference approach since the square roots x(1) = root r(1), x(2) = root r(2) allow the boundaries to be much further out. The non-linearity of the x-grid also leads to a finer description near the nucleus and a coarser one further out thus resulting in a saving of grid points. Also, in contrast to the usual variable r(12), we have instead used mu which is an independent variable. This simplifies the mathematics and numerical treatments. Several different grids can naturally run completely independent of each other thus making parallel computations trivial. From several grid results the physical property of interest is extrapolated to continuum space. The extrapolations are made in a matlab m-script where all computations can be made symbolically so the loss of decimal figures are minimized during this process. The computer code, including correlation effects and mass polarization, is highly optimized and deals with either triangular or quadratic domains in (x(1), x(2)). Restrictions: The amount of CPU time may become unreasonable for states needing boundary conditions very far beyond the origin. Also if the condition number of the corresponding Hamiltonian matrix is very high, the number of iterations will grow. The use of double precision computations also puts a limit on the accuracy of extrapolated results to about 6-7 decimal figures. Unusual features: The numerical solver is based on a particle method presented in [20, 21, 23]. In the Appendix we provide specific details of dealing with eigenvalue problems. The program uses a 64 bit environment (Linux 64bit). Parallel runs can be made conveniently through a simple bash script. Additional comments: The discretized wavefunction is complete on every given grid. New interactions can therefore conveniently be added to the Hamiltonian without the need to seek for an appropriate basis set. Running time: Given a modern CPU such as Intel core i5 and that the outer boundary conditions of r(1) and r(2) is limited to, say 16 atomic units, the total CPU time of totally 10 grids of a serial run is typically limited to a few minutes. One can then expect about 6-7 correct figures in the extrapolated eigenvalue. A single grid of say h(1) = h(2) = h(3) = 1/16 converges in less than 1 s (with an error in the eigenvalue of about 1 percent). Parallel runs are possible and can further minimize CPU times for more demanding tasks. References: [20] S. Edvardsson, M. Gulliksson, and J. Persson.). Appl. Mech. ASME, 79 (2012) 021012. [21] S. Edvardsson, M. Neuman, P Edstrom, and H. Olin. Comp. Phys. Commun. 197 (2015) 169. [23] M. Neuman, S. Edvardsson, P. Edstrom, Opt. Lett. 40 (2015) 4325.

  • 2.
    Edvardsson, Sverker
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences.
    Neuman, Magnus
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences.
    Edström, Per
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences.
    Olin, Håkan
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences.
    Solving equations through particle dynamics2015In: Computer Physics Communications, ISSN 0010-4655, E-ISSN 1879-2944, Vol. 197, p. 169-181Article in journal (Refereed)
    Abstract [en]

    The present work evaluates a recently developed particle method (DFPM). The basic idea behind this method is to utilize a Newtonian system of interacting particles that through dissipation solves mathematical problems. We find that this second order dynamical system results in an algorithm that is among the best methods known. The present work studies large systems of linear equations. Of special interest is the wide eigenvalue spectrum. This case is common as the discretization of the continuous problem becomes dense. The convergence rate of DFPM is shown to be in parity with that of the conjugate gradient method, both analytically and through numerical examples. However, an advantage with DFPM is that it is cheaper per iteration. Another advantage is that it is not restricted to symmetric matrices only, as is the case for the conjugate gradient method. The convergence properties of DFPM are shown to be superior to the closely related approach utilizing only a first order dynamical system, and also to several other iterative methods in numerical linear algebra. The performance properties are understood and optimized by taking advantage of critically damped oscillators in classical mechanics. Just as in the case of the conjugate gradient method, a limitation is that all eigenvalues (spring constants) are required to be of the same sign. DFPM has no other limitation such as matrix structure or a spectral radius as is common among iterative methods. Examples are provided to test the particle algorithm’s merits and also various performance comparisons with existent numerical algorithms are provided.

  • 3.
    Edvardsson, Sverker
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Information Technology and Media.
    Åberg, Daniel
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Information Technology and Media.
    An atomic program for energy levels of equivalent electrons: lanthanides and actinides2001In: Computer Physics Communications, ISSN 0010-4655, E-ISSN 1879-2944, Vol. 133, no 2/3, p. 396-406Article in journal (Refereed)
    Abstract [en]

    A program written in C is presented to carry out brute force calculations in order to derive energy levels for an equivalent electronic configuration. Relativistic effects are partly neglected except for the spin-orbit interaction. Since the main relativistic effects are indirect, i.e. causing a contraction of the core which in turn causes the outer shells to expand, they are included to a high degree through the use of appropriate Slater integrals. The program is especially useful for primarily unfilled f-shells of the rare-earth or actinide ions. Modifications of the program to include spin−spin, spin−other orbit, Breit interaction etc. is straight forward. The program is also general in the sense that there is no need to find out or generate any Racah coefficients of fractional parentage. The complete energy matrix is diagonalized with all operators interacting simultaneously thus allowing mixing of all quantum numbers. This result in all energy eigenvalues and eigenvectors that in turn for example are partly responsible for the polarized dipole, quadrupole, … transitions within the unfilled shell. Free ion configuration interaction is accounted for through the use of standard CI operators. The Stark splitting can be studied via the standard crystal field Hamiltonian. Magnetic field influence on the energy levels may also be studied.

  • 4.
    Edvardsson, Sverker
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Information Technology and Media.
    Åberg, Daniel
    Uddholm, Per
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Information Technology and Media.
    A program for accurate solutions of two-electron atoms2004In: Computer Physics Communications, ISSN 0010-4655, E-ISSN 1879-2944, Vol. 165, no 3, p. 260-270Article in journal (Refereed)
    Abstract [en]

    We present a comprehensible computer program capable of treating non-relativistic ground and excited states for a twoelectron atom having infinite nuclear mass. An iterative approach based on the implicitly restarted Arnoldi method (IRAM) is employed. The Hamiltonian matrix is never explicitly computed. Instead the action of the Hamiltonian operator on discrete pair functions is implemented. The finite difference method is applied and subsequent extrapolations gives the continuous grid result. The program is written in C and is highly optimized. All computations are made in double precision. Despite this relatively low degree of floating point precision (48 digits are not uncommon), the accuracy in the results can reach about 10 significant figures. Both serial and parallel versions are provided. The parallel program is particularly suitable for shared memory machines such as the Sun Starcat series. The serial version is simple to compile and should run on any platform.

  • 5.
    Wiklund, Hanna
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
    Lindström, S. B.
    Department of Fibre and Polymer Technology, Royal Institute of Technology, SE-100 44 Stockholm.
    Uesaka, Tetsu
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
    Boundary condition considerations in lattice Boltzmann formulations of wetting binary fluids2011In: Computer Physics Communications, ISSN 0010-4655, E-ISSN 1879-2944, Vol. 182, no 10, p. 2192-2200Article in journal (Refereed)
    Abstract [en]

    We propose a new lattice Boltzmann numerical scheme for binary-fluid surface interactions. The new scheme combines the existing binary free energy lattice Boltzmann method [Swift et al., Phys. Rev. E 54 (1996)] and a new wetting boundary condition for diffuse interface methods in order to eliminate spurious variations in the order parameter at solid surfaces. We use a cubic form for the surface free energy density and also take into account the contribution from free energy in the volume when discretizing the wetting boundary condition. This allows us to eliminate the spurious variation in the order parameter seen in previous implementations. With the new scheme a larger range of equilibrium contact angles are possible to reproduce and capillary intrusion can be simulated at higher accuracy at lower resolution. © 2011 Elsevier B.V. All rights reserved.

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