Time-splitting pseudo-spectral domain decomposition method for the soliton solutions of the one- and multi-dimensional nonlinear Schrödinger equations

In this paper, we study the simulation of nonlinear Schrödinger equation in one, two and three dimensions. The proposed method is based on a time-splitting method that decomposes the original problem into two parts, a linear equation and a nonlinear equation. The linear equation in one dimension is approximated with the Chebyshev pseudo-spectral collocation method in space variable and the Crank–Nicolson method in time; while the nonlinear equation with constant coefficients can be solved exactly. As the goal of the present paper is to study the nonlinear Schrödinger equation in the large finite domain, we propose a domain decomposition method. In comparison with the single-domain, the multi-domain methods can produce a sparse differentiation matrix with fewer memory space and less computations. In this study, we choose an overlapping multi-domain scheme. By applying the alternating direction implicit technique, we extend this efficient method to solve the nonlinear Schrödinger equation both in two and three dimensions, while for the solution at each time step, it only needs to solve a sequence of linear partial differential equations in one dimension, respectively. Several examples for one- and multi-dimensional nonlinear Schrödinger equations are presented to demonstrate high accuracy and capability of the proposed method. Some numerical experiments are reported which show that this scheme preserves the conservation laws of charge and energy.     

A Truncation Method for Computing Walsh Transforms with Applications to Image Processing

We present a method called the Truncation method for computing Walsh-Hadamard transforms of one- and two-dimensional data. In one dimension, the method uses binary trees as a basis for representing the data and computing the transform. In two dimensions, the method uses quadtrees (pyramids), adaptive quad-trees, or binary trees as a basis. We analyze the storage and time complexity of this method in worst and general cases. The results show that the Truncation method degenerates to the Fast Walsh Transform (FWT) in the worst case, while the Truncation method is faster than the Fast Walsh Transform when there is coherence in the input data, as will typically be the case for image data. In one dimension, the performance of the Truncation method for N data samples is between O(N) and O(N log₂N), and it is between O(N²) and O(N² log₂N) in two dimensions. Practical results on several images are presented to show that both the expected and actual overall times taken to compute Walsh transforms using the Truncation method are less than those required by a similar implementation of the FWT method.     

Solving the vibrational Schrödinger equation on an arbitrary multidimensional potential energy surface by the finite element method

A computational protocol has been developed to solve the bounded vibrational Schrödinger equation for up to three coupled coordinates on any given effective potential energy surface (PES). The dynamic Wilson G-matrix is evaluated from the discrete PES calculations, allowing the PES to be parametrized in terms of any complete, minimal set of coordinates, whether orthogonal or non-orthogonal. The partial differential equation is solved using the finite element method (FEM), to take advantage of its localized basis set structure and intrinsic scalability to multiple dimensions. A mixed programming paradigm takes advantage of existing libraries for constructing the FEM basis and carrying out the linear algebra. Results are presented from a series of calculations confirming the flexibility, accuracy, and efficiency of the protocol, including tests on FHF⁻, picolinic acid N-oxide, trans-stilbene, a generalized proton transfer system, and selected model systems.     

A Fast Spectral Subtractional Solver for Elliptic Equations

The paper presents a fast subtractional spectral algorithm for the solution of the Poisson equation and the Helmholtz equation which does not require an extension of the original domain. It takes O(N ² log N) operations, where N is the number of collocation points in each direction. The method is based on the eigenfunction expansion of the right hand side with integration and the successive solution of the corresponding homogeneous equation using Modified Fourier Method. Both the right hand side and the boundary conditions are not assumed to have any periodicity properties. This algorithm is used as a preconditioner for the iterative solution of elliptic equations with non-constant coefficients. The procedure enjoys the following properties: fast convergence and high accuracy even when the computation employs a small number of collocation points. We also apply the basic solver to the solution of the Poisson equation in complex geometries.     

Quasilinearization approach to quantum mechanics

The quasilinearization method (QLM) of solving nonlinear differential equations is applied to the quantum mechanics by casting the Schrödinger equation in the nonlinear Riccati form. The method, whose mathematical basis in physics was discussed recently by one of the present authors (VBM), approaches the solution of a nonlinear differential equation by approximating the nonlinear terms by a sequence of the linear ones, and is not based on the existence of some kind of a small parameter. It is shown that the quasilinearization method gives excellent results when applied to computation of ground and excited bound state energies and wave functions for a variety of the potentials in quantum mechanics most of which are not treatable with the help of the perturbation theory or the 1/N expansion scheme. The convergence of the QLM expansion of both energies and wave functions for all states is very fast and already the first few iterations yield extremely precise results. The precision of the wave function is typically only one digit inferior to that of the energy. In addition it is verified that the QLM approximations, unlike the asymptotic series in the perturbation theory and the 1/N expansions are not divergent at higher orders.     

A compact split-step finite difference method for solving the nonlinear Schrödinger equations with constant and variable coefficients

We propose a compact split-step finite difference method to solve the nonlinear Schrödinger equations with constant and variable coefficients. This method improves the accuracy of split-step finite difference method by introducing a compact scheme for discretization of space variable while this improvement does not reduce the stability range and does not increase the computational cost. This method also preserves some conservation laws. Numerical tests are presented to confirm the theoretical results for the new numerical method by using the cubic nonlinear Schrödinger equation with constant and variable coefficients and Gross-Pitaevskii equation.     

Quasilinearization approach to computations with singular potentials

We pioneered the application of the quasilinearization method (QLM) to the numerical solution of the Schrödinger equation with singular potentials. The spiked harmonic oscillator r²+λr−α is chosen as the simplest example of such potential. The QLM has been suggested recently for solving the Schrödinger equation after conversion into the nonlinear Riccati form. In the quasilinearization approach the nonlinear differential equation is treated by approximating the nonlinear terms by a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. The choice of zero iteration is based on general features of solutions near the boundaries.We show that the energies of bound state levels in the spiked harmonic oscillator potential which are notoriously difficult to compute for small couplings λ, are easily calculated with the help of QLM for any λ and α with accuracy of twenty significant figures.     

A new kind of discretization scheme for solving a two-dimensional time-independent Schrödinger equation

In this paper we present a new kind of discretization scheme for solving a two-dimensional time-independent Schrödinger equation. The scheme uses a symmetrical multi-point difference formula to represent the partial differentials of the two-dimensional variables, which can improve the accuracy of the numerical solutions to the order of Δx2Nq+2 when a (2Nq+1)-point formula is used for any positive integer Nq with Δx=Δy, while Nq=1 equivalent to the traditional scheme. On the other hand, the new scheme keeps the same form of the traditional matrix equation so that the standard algebraic eigenvalue algorithm with a real, symmetric, large sparse matrix is still applicable. Therefore, for the same dimension, only a little more CPU time than the traditional one should be used for diagonalizing the matrix. The numerical examples of the two-dimensional harmonic oscillator and the two-dimensional Henon-Heiles potential demonstrate that by using the new method, the error in the numerical solutions can be reduced steadily and extensively through the increase of Nq, which is more efficient than the traditional methods through the decrease of the step size.     

A method for solving the molecular Schrödinger equation in Cartesian coordinates via angular momentum projection operators

A method for solving the Schrödinger equation of N-atom molecules in 3N−3 Cartesian coordinates usually defined by Jacobi vectors is presented. The separation and conservation of the total angular momentum are obtained not by transforming the Hamiltonian in internal curvilinear coordinates but instead, by keeping the Cartesian formulation of the Hamiltonian operator and projecting the initial wavefunction onto the proper irreducible representation angular momentum subspace. The increased number of degrees of freedom from 3N−6 to 3N−3, compared to previous methods for solving the Schrödinger equation, is compensated by the simplicity of the kinetic energy operator and its finite difference representations which result in sparse Hamiltonian matrices. A parallel code in Fortran 95 has been developed and tested for model potentials of harmonic oscillators. Moreover, we compare data obtained for the three-dimensional hydrogen molecule and the six-dimensional water molecule with results from the literature. The availability of large clusters of computers with hundreds of CPUs and GBytes of memory, as well as the rapid development of distributed (Grid) computing, make the proposed method, which is unequivocally highly demanding in memory and computer time, attractive for studying Quantum Molecular Dynamics.     

Optimal Elections in Labeled Hypercubes

We study the message complexity of theElectionProblem in hypercube networks, when the processors have a “Sense of Direction,” i.e., the capability to distinguish between adjacent communication links according to some globally consistent scheme. We present two models of Sense of Direction, which differ regarding the way the labeling of the links in the network is done: either by matching based on dimensions or by distance along a Hamiltonian cycle. In the dimension model, we give an optimal linear algorithm which uses the natural dimensional labeling of the communication links. We prove that, in the distance-based case, the graph symmetry of the hypercube is broken and, thus, the leader Election does not require a global maximum-finding algorithm:O(1) messages suffice to select the leader, whereas the Θ(N) messages are required only for the final broadcasting. Finally, we study the communication cost of changing one orientation labeling to the other and prove thatO(N) messages suffice.     

Space–time adaptive finite difference method for European multi-asset options

The multi-dimensional Black–Scholes equation is solved numerically for a European call basket option using a priori–a posteriori error estimates. The equation is discretized by a finite difference method on a Cartesian grid. The grid is adjusted dynamically in space and time to satisfy a bound on the global error. The discretization errors in each time step are estimated and weighted by the solution of the adjoint problem. Bounds on the local errors and the adjoint solution are obtained by the maximum principle for parabolic equations. Comparisons are made with Monte Carlo and quasi-Monte Carlo methods in one dimension, and the performance of the method is illustrated by examples in one, two, and three dimensions.     

MOMCON: A spectral code for obtaining three-dimensional magnetohydrodynamic equilibria

A new code, MOMCON (spectral moments code with constraints), is described that computes three-dimensional ideal magnetohydrodynamic (MHD) equilibria in a fixed toroidal domain using a Fourier expansion for the inverse coordinates (R, Z) representing nested magnetic surfaces. A set of nonlinear coupled ordinary differential equations for the spectral coefficients of (R, Z) is solved using an accelerated steepest descent method. A stream function, λ, is introduced to improve the mode convergence properties of the Fourier series for R and Z. The convergence rate of the R - Z spectra is optimized on each flux surface by solving nonlinear constraint equations relating the m ≥ 2 spectral coefficients of R and Z.     

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.     

A conservative exponential time differencing method for the nonlinear cubic Schrödinger equation

A mass and energy conservative exponential time differencing scheme using the method of lines is proposed for the numerical solution of a certain family of first-order time-dependent PDEs. The resulting nonlinear system is solved with an unconditionally stable modified predictor–corrector method using a second-order explicit scheme. The efficiency of the method introduced is analyzed and discussed by applying it to the nonlinear cubic Schrödinger equation. The results arising from the experiments for the single, the double soliton waves and the system of two Schrödinger equations are compared with relevant known ones.     

Discretization of elliptic differential equations on curvilinear bounded domains with sparse grids

Elliptic differential equations can be discretized with bilinear finite elements. Using sparse grids instead of full grids, the dimension of the finite element space for the 2D problem reduces fromO(N ²) toO (N logN) while the approximation properties are nearly the same for smooth functions. A method is presented which discretizes elliptic differential equations on curvilinear bounded domains with adaptive sparse grids. The grid is generated by a transformation of the domain. This method has the same behaviour of convergence like the sparse grid discretization on the unit square.     

Use of a double Fourier series for three-dimensional shape representation

The representation of three-dimensional star-shaped objects by the double Fourier series (DFS) coefficients of their boundary function is considered. An analogue of the convolution theorem for a DFS on a sphere is developed. It is then used to calculate the moments of an object directly from the DFS coefficients, without an intermediate reconstruction step. The complexity of computing the moments from the DFS coefficients is O(N ² log N), where N is the maximum order of coefficients retained in the expansion, while the complexity of computing the moments from the spherical harmonic representation is O(N ² log ² N). It is shown that under sufficient conditions, the moments and surface area corresponding to the truncated DFS converge to the true moments and area of an object. A new kind of DFS—the double Fourier sine series—is proposed which has better convergence properties than the previously used kinds and spherical harmonics in the case of objects with a sharp point above the pole of the spherical domain.     

An algebraic method for Schrödinger equations in quaternionic quantum mechanics

In the study of theory and numerical computations of quaternionic quantum mechanics and quantum chemistry, one of the most important tasks is to solve the Schrödinger equation with A an anti-self-adjoint real quaternion matrix, and |f〉 an eigenstate to A. The quaternionic Schrödinger equation plays an important role in quaternionic quantum mechanics, and it is known that the study of the quaternionic Schrödinger equation is reduced to the study of quaternionic eigen-equation Aα=αλ with A an anti-self-adjoint real quaternion matrix (time-independent). This paper, by means of complex representation of quaternion matrices, introduces concepts of norms of quaternion matrices, studies the problems of quaternionic Least Squares eigenproblem, and give a practical algebraic technique of computing approximate eigenvalues and eigenvectors of a quaternion matrix in quaternionic quantum mechanics.     

An improved algorithm for Boolean matrix multiplication

A new algorithm for computing the product of two arbitraryN×N Boolean matrices is presented. The algorithm requiresO (N ³/logN) bit operations and onlyO(N logN) bits of additional storage. This represents an improvement on the Four Russians' method which requires the same number of operations but usesO(N ³/logN) bits of additional storage.     

Fourth-order algorithms for solving local Schrödinger equations in a strong magnetic field

We describe an efficient numerical method for solving eigenvalue problems associated with the one-body Schrödinger equation or the Kohn-Sham equations in an arbitrarily strong uniform external magnetic field. The eigenvalue problem is solved in real space by using a fourth order, forward factorization of the evolution operator e−εH, which is significantly more efficient than conventional second-order algorithms. In particular, the magnetic field is solved exactly by the decomposition process. The algorithm is applicable to any external potential, in addition to the magnetic field. We envision its primary application in the area of electronic structure calculations of quantum dots.