Accurate computations for the elastic scattering phase-shift problem

A new hybrid sixth algebraic order two-step method is developed for computing elastic scattering phase-shifts of the one-dimensional Schrödinger equation. Based on this new method and on the method developed recently by Simos and Tougelidis [Computers Chem. 20, 397–401 (1996)], we obtain a new variable-step procedure for the numerical integration of the Schrödinger equation. Numerical results obtained for the integration of the phase-shift problem for the well-known case of the Lenard-Jones potential show that this new method is better than other finite difference methods.     

A dissipative exponentially-fitted method for the numerical solution of the Schrödinger equation and related problems

In this paper a dissipative exponentially-fitted method for the numerical integration of the Schrödinger equation and related problems is developed. The method is called dissipative since is a nonsymmetric multistep method. An application to the the resonance problem of the radial Schrödinger equation and to other well known related problems indicates that the new method is more efficient than the corresponding classical dissipative method and other well known methods. Based on the new method and the method of Raptis and Cash a new variable-step method is obtained. The application of the new variable-step method to the coupled differential equations arising from the Schrödinger equation indicates the power of the new approach.     

Dissipative exponentially-fitted methods for the numerical solution of the Schrödinger equation

The first dissipative exponentially fitted method for the numerical integration of the Schr?dinger equation is developed in this paper. The technique presented is a nonsymmetric multistep (dissipative) method. An application to the bound-states problem and the resonance problem of the radial Schr?dinger equation indicates that the new method is more efficient than the classical dissipative method and other well-known methods. Based on the new method and the method of Raptis and Allison (Comput. Phys. Commun. 14 (1978) 1-5) a new variable-step method is obtained. The application of the new variable-step method to the coupled differential equations arising from the Schr?dinger equation indicates the power of the new approach.     

On variable-step methods for the numerical solution of Schrödinger equation and related problems

In this paper we present a review for the construction of variable-step methods for the numerical integration of the Schr?dinger equation. Phase-lag and stability are investigated. The methods are variable-step because of a simple natural error control mechanism. Numerical results obtained for coupled differential equations arising from the Schr?dinger equation and for the wave equation show the validity of the approach presented.     

A high order method for the numerical integration of the one-dimensional Schrödinger equation

A sixth order method is developed for the approximate numerical integration of the one-dimensional Schrödinger equation. Numerical results obtained for the integration of both the eigenvalue and the phase shift problems show that this new method is generally superior to the widely used Numerov method.     

A program for perspective views of three-dimensional surfaces

A numerical method is proposed for solving linear differential equations of second order without first derivatives. The new method is superior to de Vogelaere's for this class of equations, and for non-linear equations it becomes an implicit extension of de Vogelaere's method. The global truncation error at a fixed steplength h is bounded by a term of order h⁴, and the interval of absolute stability is [?2.4, 0]. The work of Coleman and Mohamed (1978) is readily adapted to provide truncation error estimates which can be used for automatic error control. It is suggested that the new method should be used in preference to de Vogelaere's for linear equations, and in particular to solve the radial Schrödinger equation. the radial Schrödinger equation.     

A numerical method for solving a coupled channel Schrödinger equation

An approximation method involving spherical delta functions is presented for the solving of coupled channel differential equations, in particular the Schrödinger equation. A specific example is worked out in detail.     

A generalized Suzuki–Trotter type method in optimal control of coupled Schrödinger equations

A generalized Suzuki–Trotter (GST) method for the solution of an optimal control problem for quantum molecular systems is presented in this work. The control of such systems gives rise to a minimization problem with constraints given by a system of coupled Schrödinger equations. The computational bottleneck of the corresponding minimization methods is the solution of time-dependent Schrödinger equations. To solve the Schrödinger equations we use the GST framework to obtain an explicit polynomial approximation of the matrix exponential function. The GST method almost exclusively uses the action of the Hamiltonian and is therefore efficient and easy to implement for a variety of quantum systems. Following a first discretize, then optimize approach we derive the correct discrete representation of the gradient and the Hessian. The derivatives can naturally be expressed in the GST framework and can therefore be efficiently computed. By recomputing the solutions of the Schrödinger equations instead of saving the whole time evolution, we are able to significantly reduce the memory requirements of the method at the cost of additional computations. This makes first and second order optimization methods viable for large scale problems. In numerical experiments we compare the performance of different first and second order optimization methods using the GST method. We observe fast local convergence of second order methods.     

Numerov-type methods with minimal phase-lag for the numerical integration of the one-dimensional Schrödinger equation

Three Numerov-type methods with phase-lag of order eight and ten are developed for the numerical integration of the one-dimensional Schrödinger equation. One has a large interval of periodicity and the other two areP-stable. Extensive numerical testing on the resonance problem indicates that these new methods are generally more accurate than other previously developed finite difference methods for this problem.     

New schemes for the coupled nonlinear Schrödinger equation

In this paper, we present three new schemes for the coupled nonlinear Schrödinger equation. The three new schemes are multi-symplectic schemes that preserve the intrinsic geometry property of the equation. The three new schemes are also semi-explicit in the sense that they need not solve linear algebraic equations every time-step, which is usually the most expensive in numerical simulation of partial differential equations. Many numerical experiments on collisions of solitons are presented to show the efficiency of the new multi-symplectic schemes.     

P-stable eighth algebraic order methods for the numerical solution of the Schr?dinger equation.

A P-stable method of algebraic order eight for the approximate numerical integration of the Schr?dinger equation is developed in this paper. Since the method is P-stable (i.e. its interval of periodicity is equal to (0, infinity)), large step sizes for the numerical integration can be used. Based on this new method and on a sixth algebraic order P-stable method developed by Simos (Phys. Scripta 55 (1997) 644-650), a new variable step method is obtained. Numerical results presented for the phase-shift problem of the radial Schr?dinger equation and for the coupled differential equations arising from the Schr?dinger equation show the efficiency of the developed method.     

An efficient split-step compact finite difference method for the coupled Gross–Pitaevskii equations

ABSTRACT

In this study, we propose an efficient split-step compact finite difference (SSCFD) method for computing the coupled Gross–Pitaevskii (CGP) equations. The coupled equations are divided into two parts, nonlinear subproblems and linear ones. Commonly, the nonlinear subproblems could be integrated directly and accurately, but it fails when the time-dependent potential cannot be integrated exactly. In this case, the midpoint and trapezoidal rules are applied approximately. At the same time, the split order is not reduced. For the linear ones, compact finite difference cannot be designed directly. To circumvent this problem, a linear transformation is introduced to decouple the system, which can make the split-step method be used again. Additionally, the proposed SSCFD method also holds for the coupled nonlinear Schrödinger (CNLS) system with time-dependent potential. Finally, numerical experiments for CGP equations and CNLS equations are well simulated, conservative properties and convergence rates are demonstrated as well. It is shown from the numerical tests that the present method is efficient and reliable.     

A numerov-type method for computing eigenvalues and resonances of the radial Schrödinger equation

A two-step method is developed for computing eigenvalues and resonances of the radial Schrödinger equation. Numerical results obtained for the integration of the eigenvalue and the resonance problems for several potentials show that this new method is better than other similar methods.     

A variable step method for the numerical integration of the one-dimensional Schrödinger equation

Most numerical methods which have been proposed for the approximate integration of the one-dimensional Schrödinger equation use a fixed step length of integration. Such an approach can of course result in gross inefficiency since the small step length which must normally be used in the initial part of the range of integration to obtain the desired accuracy must then be used throughout the integration. In this paper we consider the method of embedding, which is widely used with explicit Runge-Kutta methods for the solution of first order initial value problems, for use with the special formulae used to integrate the Schrödinger equation. By adopting this technique we have available at each step an estimate of the local truncation error and this estimate can be used to automatically control the step length of integration. Also considered is the problem of estimating the global truncation error at the end of the range of integration. The power of the approaches considered is illustrated by means of some numerical examples.     

Domain Decomposition Algorithms for Two Dimensional Linear Schrödinger Equation

This paper deals with two domain decomposition methods for two dimensional linear Schrödinger equation, the Schwarz waveform relaxation method and the domain decomposition in space method. After presenting the classical algorithms, we propose a new algorithm for the Schrödinger equation with constant potential and a preconditioned algorithm for the general Schrödinger equation. These algorithms are then studied numerically. The numerical experiments show that the new algorithms can improve the convergence rate and reduce the computation time. Besides of the traditional Robin transmission condition, we also propose to use a newly constructed absorbing condition as the transmission condition.     

Variable preconditioning in complex Hilbert space and its application to the nonlinear Schrödinger equation

An iterative method is developed for nonlinear equations in complex Hilbert spaces, extending the method of variable preconditioning defined earlier in real spaces. We derive convergence of our method. The motivating example for this extension is the time-dependent nonlinear Schrödinger equation, where we use our iteration for the time discretization of the problem and test it numerically.     

Coefficients perturbation methods for higher-order differential equations

In this paper, we develop a general approach for estimating and bounding the error committed when higher-order ordinary differential equations (ODEs) are approximated by means of the coefficients perturbation methods. This class of methods was specially devised for the solution of Schrödinger equation by Ixaru in 1984. The basic principle of perturbation methods is to find the exact solution of an approximation problem obtained from the original one by perturbing the coefficients of the ODE, as well as any supplementary condition associated to it. Recently, the first author obtained practical formulae for calculating tight error bounds for the perturbation methods when this technique is applied to second-order ODEs. This paper extends those results to the case of differential equations of arbitrary order, subjected to some specified initial or boundary conditions. The results of this paper apply to any perturbation-based numerical technique such as the segmented Tau method, piecewise collocation, Constant and Linear perturbation. We will focus on the Tau method and present numerical examples that illustrate the accuracy of our results.     

Restrictive Padé Approximation for the Solution of the SchröDinger Equation

The problem of solving the Schrödinger equation by a method related to the restrictive Padé approximation is considered. It yields more accurate results. The complex tridiagonal system which arises from the finite difference discretization of the considered equation is solved by Evans-Roomi [1] method. The restrictive Padé approach is applied successfully for the one and two dimensional Schrödinger equations. It is shown by numerical examples that it is more efficient and gives faster results compared with classical finite difference methods.     

A generator of dissipative methods for the numerical solution of the Schrödinger equation

In this paper a generator of hybrid methods with minimal phase-lag is developed for the numerical solution of the Schrödinger equation and related problems. The generator's methods are dissipative and are of eighth algebraic order. In order to have minimal phase-lag with the new methods, their coefficients are determined automatically. Numerical results obtained by their application to some well known problems with periodic or oscillating solutions and to the coupled differential equations of the Schrödinger type indicate the efficiency of these new methods.     

The WKB Local Discontinuous Galerkin Method for the Simulation of Schrödinger Equation in a Resonant Tunneling Diode

In this paper, we develop a multiscale local discontinuous Galerkin (LDG) method to simulate the one-dimensional stationary Schrödinger-Poisson problem. The stationary Schrödinger equation is discretized by the WKB local discontinuous Galerkin (WKB-LDG) method, and the Poisson potential equation is discretized by the minimal dissipation LDG (MD-LDG) method. The WKB-LDG method we propose provides a significant reduction of both the computational cost and memory in solving the Schrödinger equation. Compared with traditional continuous finite element Galerkin methodology, the WKB-LDG method has the advantages of the DG methods including their flexibility in h-p adaptivity and allowance of complete discontinuity at element interfaces. Although not addressed in this paper, a major advantage of the WKB-LDG method is its feasibility for two-dimensional devices.