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1.
Katharina Kormann 《Communications In Computational Physics》2016,20(1):60-85
In this paper, we present a discretization of the time-dependent Schrödinger
equation based on a Magnus-Lanczos time integrator and high-order Gauss-Lobatto
finite elements in space. A truncated Galerkin orthogonality is used to obtain duality-based
a posteriori error estimates that address the temporal and the spatial error separately.
Based on this theory, a space-time adaptive solver for the Schrödinger equation
is devised. An efficient matrix-free implementation of the differential operator, suited
for spectral elements, is used to enable computations for realistic configurations. We
demonstrate the performance of the algorithm for the example of matter-field interaction. 相似文献
2.
Hua Guan Yandong Jiao Ju Liu & Yifa Tang 《Communications In Computational Physics》2009,6(3):639-654
By performing a particular spatial discretization to the nonlinear
Schrödinger equation (NLSE), we obtain a non-integrable Hamiltonian system which
can be decomposed into three integrable parts (L-L-N splitting). We integrate each part
by calculating its phase flow, and develop explicit symplectic integrators of different
orders for the original Hamiltonian by composing the phase flows. A 2nd-order reversible
constructed symplectic scheme is employed to simulate solitons motion and
invariants behavior of the NLSE. The simulation results are compared with a 3rd-order
non-symplectic implicit Runge-Kutta method, and the convergence of the formal energy
of this symplectic integrator is also verified. The numerical results indicate that
the explicit symplectic scheme obtained via L-L-N splitting is an effective numerical
tool for solving the NLSE. 相似文献
3.
We derive a perfectly matched layer (PML) for the Schrödinger equation using
a modal ansatz. We derive approximate error formulas for the modeling error from
the outer boundary of the PML and the error from the discretization in the layer and
show how to choose layer parameters so that these errors are matched and optimal
performance of the PML is obtained. Numerical computations in 1D and 2D demonstrate
that the optimized PML works efficiently at a prescribed accuracy for the zero
potential case, with a layer of width less than a third of the de Broglie wavelength
corresponding to the dominating frequency. 相似文献
4.
Jiwei Zhang Zhizhong Sun Xiaonan Wu & Desheng Wang 《Communications In Computational Physics》2011,10(3):742-766
The paper is concerned with the numerical solution of Schrödinger equations
on an unbounded spatial domain. High-order absorbing boundary conditions
for one-dimensional domain are derived, and the stability of the reduced initial boundary
value problem in the computational interval is proved by energy estimate. Then a
second order finite difference scheme is proposed, and the convergence of the scheme
is established as well. Finally, numerical examples are reported to confirm our error
estimates of the numerical methods. 相似文献
5.
Semi-Eulerian and High Order Gaussian Beam Methods for the Schrödinger Equation in the Semiclassical Regime 下载免费PDF全文
A novel Eulerian Gaussian beam method was developed in [8] to compute
the Schrödinger equation efficiently in the semiclassical regime. In this paper, we introduce
an efficient semi-Eulerian implementation of this method. The new algorithm
inherits the essence of the Eulerian Gaussian beam method where the Hessian is computed
through the derivatives of the complexified level set functions instead of solving
the dynamic ray tracing equation. The difference lies in that, we solve the ray tracing
equations to determine the centers of the beams and then compute quantities of interests
only around these centers. This yields effectively a local level set implementation,
and the beam summation can be carried out on the initial physical space instead of the
phase plane. As a consequence, it reduces the computational cost and also avoids the
delicate issue of beam summation around the caustics in the Eulerian Gaussian beam
method. Moreover, the semi-Eulerian Gaussian beam method can be easily generalized
to higher order Gaussian beam methods, which is the topic of the second part
of this paper. Several numerical examples are provided to verify the accuracy and
efficiency of both the first order and higher order semi-Eulerian methods. 相似文献
6.
A Preconditioned Iterative Solver for the Scattering Solutions of the Schrödinger Equation 下载免费PDF全文
Hisham bin Zubair Bram Reps & Wim Vanroose 《Communications In Computational Physics》2012,11(2):415-434
The Schrödinger equation defines the dynamics of quantum particles which has been an area of unabated interest in physics. We demonstrate how simple transformations of the Schrödinger equation leads to a coupled linear system, whereby each diagonal block is a high frequency Helmholtz problem. Based on this model, we derive indefinite Helmholtz model problems with strongly varying wavenumbers. We employ the iterative approach for their solution. In particular, we develop a preconditioner that has its spectrum restricted to a quadrant (of the complex plane) thereby making it easily invertible by multigrid methods with standard components. This multigrid preconditioner is used in conjunction with suitable Krylov-subspace methods for solving the indefinite Helmholtz model problems. The aim of this study is to report the feasibility of this preconditioner for the model problems. We compare this idea with the other prevalent preconditioning ideas, and discuss its merits. Results of numerical experiments are presented, which complement the proposed ideas, and show that this preconditioner may be used in an automatic setting. 相似文献
7.
Analysis and Efficient Solution of Stationary Schrödinger Equation Governing Electronic States of Quantum Dots and Rings in Magnetic Field 下载免费PDF全文
In this work the one-band effective Hamiltonian governing the electronic states of a quantum dot/ring in a homogenous magnetic field is used to derive a pair/quadruple of nonlinear eigenvalue problems corresponding to different spin orientations and in case of rotational symmetry additionally to quantum number ±ℓ. We show, that each of those pair/quadruple of nonlinear problems allows for the min-max characterization of its eigenvalues under certain conditions, which are satisfied for our examples and the common InAs/GaAs heterojunction. Exploiting the minmax property we devise efficient iterative projection methods simultaneously handling the pair/quadruple of nonlinear problems and thereby saving up to 40% of the computational time as compared to the nonlinear Arnoldi method applied to each of the problems separately. 相似文献
8.
Shanshan Jiang Lijin Wang & Jialin Hong 《Communications In Computational Physics》2013,14(2):393-411
In this paper we propose stochastic multi-symplectic conservation law for stochastic Hamiltonian partial differential equations, and develop a stochastic multi-symplectic method for numerically solving a kind of stochastic nonlinear Schrödinger equations. It is shown that the stochastic multi-symplectic method preserves the multi-symplectic structure, the discrete charge conservation law, and deduces the recurrence relation of the discrete energy. Numerical experiments are performed to verify the good behaviors of the stochastic multi-symplectic method in cases of both solitary wave and collision. 相似文献
9.
Excitonic Eigenstates of Disordered Semiconductor Quantum Wires: Adaptive Wavelet Computation of Eigenvalues for the Electron-Hole Schrödinger Equation 下载免费PDF全文
Christian Mollet Angela Kunoth & Torsten Meier 《Communications In Computational Physics》2013,14(1):21-47
A novel adaptive approach to compute the eigenenergies and eigenfunctions of the two-particle (electron-hole) Schrödinger equation including Coulomb attraction is presented. As an example, we analyze the energetically lowest exciton state of a thin one-dimensional semiconductor quantum wire in the presence of disorder which arises from the non-smooth interface between the wire and surrounding material. The eigenvalues of the corresponding Schrödinger equation, i.e., the one-dimensional exciton Wannier equation with disorder, correspond to the energies of excitons in the quantum wire. The wavefunctions, in turn, provide information on the optical properties of the wire. We reformulate the problem of two interacting particles that both can move in one dimension as a stationary eigenvalue problem with two spacial dimensions in an appropriate weak form whose bilinear form is arranged to be symmetric, continuous, and coercive. The disorder of the wire is modelled by adding a potential in the Hamiltonian which is generated by normally distributed random numbers. The numerical solution of this problem is based on adaptive wavelets. Our scheme allows for a convergence proof of the resulting scheme together with complexity estimates. Numerical examples demonstrate the behavior of the smallest eigenvalue, the ground state energies of the exciton, together with the eigenstates depending on the strength and spatial correlation of disorder. 相似文献
10.
Computing the Ground and First Excited States of the Fractional Schrödinger Equation in an Infinite Potential Well 下载免费PDF全文
In this paper, we numerically study the ground and first excited states of the
fractional Schrödinger equation in an infinite potential well. Due to the nonlocality of
the fractional Laplacian, it is challenging to find the eigenvalues and eigenfunctions
of the fractional Schrödinger equation analytically. We first introduce a normalized
fractional gradient flow and then discretize it by a quadrature rule method in space
and the semi-implicit Euler method in time. Our numerical results suggest that the
eigenfunctions of the fractional Schrödinger equation in an infinite potential well differ
from those of the standard (non-fractional) Schrödinger equation. We find that the
strong nonlocal interactions represented by the fractional Laplacian can lead to a large
scattering of particles inside of the potential well. Compared to the ground states,
the scattering of particles in the first excited states is larger. Furthermore, boundary
layers emerge in the ground states and additionally inner layers exist in the first excited
states of the fractional nonlinear Schrödinger equation. Our simulated eigenvalues are
consistent with the lower and upper bound estimates in the literature. 相似文献
11.
Review of Feynman's Path Integral in Quantum Statistics: From the Molecular Schrödinger Equation to Kleinert's Variational Perturbation Theory 下载免费PDF全文
Kin-Yiu Wong 《Communications In Computational Physics》2014,15(4):853-894
Feynman's path integral reformulates the quantum Schrödinger differential
equation to be an integral equation. It has been being widely used to compute internuclear
quantum-statistical effects on many-body molecular systems. In this Review,
the molecular Schrödinger equation will first be introduced, together with the Born-Oppenheimer
approximation that decouples electronic and internuclear motions. Some
effective semiclassical potentials, e.g., centroid potential, which are all formulated in
terms of Feynman's path integral, will be discussed and compared. These semiclassical
potentials can be used to directly calculate the quantum canonical partition function
without individual Schrödinger's energy eigenvalues. As a result, path integrations
are conventionally performed with Monte Carlo and molecular dynamics sampling
techniques. To complement these techniques, we will examine how Kleinert's variational
perturbation (KP) theory can provide a complete theoretical foundation for developing
non-sampling/non-stochastic methods to systematically calculate centroid
potential. To enable the powerful KP theory to be practical for many-body molecular
systems, we have proposed a new path-integral method: automated integration-free
path-integral (AIF-PI) method. Due to the integration-free and computationally
inexpensive characteristics of our AIF-PI method, we have used it to perform ab initio
path-integral calculations of kinetic isotope effects on proton-transfer and RNA-related
phosphoryl-transfer chemical reactions. The computational procedure of using our
AIF-PI method, along with the features of our new centroid path-integral theory at the
minimum of the absolute-zero energy (AMAZE), are also highlighted in this review. 相似文献
12.
Efficient Grid Treatment of the Time Dependent Schrödinger Equation for Laser-Driven Molecular Dynamics 下载免费PDF全文
We present an efficient method to solve the time dependent Schrödinger equation for modeling the dynamics of diatomic molecules irradiated by intense ultrashort laser pulse without Born-Oppenheimer approximation. By introducing a variable prolate spheroidal coordinates and discrete variable representations of the Hamiltonian, we can accurately and efficiently simulate the motion of both electronic and molecular dynamics. The accuracy and convergence of this method are tested by simulating the molecular structure, photon ionization and high harmonic generation of $H_2^+$. 相似文献
13.
An Exact Absorbing Boundary Condition for the Schrödinger Equation with Sinusoidal Potentials at Infinity 下载免费PDF全文
Chunxiong Zheng 《Communications In Computational Physics》2008,3(3):641-658
In this paper we study numerical issues related to the Schrödinger equation
with sinusoidal potentials at infinity. An exact absorbing boundary condition in a form
of Dirichlet-to-Neumann mapping is derived. This boundary condition is based on an
analytical expression of the logarithmic derivative of the Floquet solution to Mathieu's
equation, which is completely new to the author's knowledge. The implementation
of this exact boundary condition is discussed, and a fast evaluation method is used to
reduce the computation burden arising from the involved half-order derivative operator. Some numerical tests are given to show the performance of the proposed absorbing
boundary conditions. 相似文献
14.
Efficient Splitting Methods Based on Modified Potentials: Numerical Integration of Linear Parabolic Problems and Imaginary Time Propagation of the Schrödinger Equation 下载免费PDF全文
Sergio Blanes Fernando Casas Cesá reo Gonzá lez & Mechthild Thalhammer 《Communications In Computational Physics》2023,33(4):937-961
We present a new family of fourth-order splitting methods with positive coefficients especially tailored for the time integration of linear parabolic problems and,in particular, for the time dependent Schrödinger equation, both in real and imaginarytime. They are based on the use of a double commutator and a modified processor, andare more efficient than other widely used schemes found in the literature. Moreover,for certain potentials, they achieve order six. Several examples in one, two and threedimensions clearly illustrate the computational advantages of the new schemes. 相似文献
15.
Fourth Order Exponential Time Differencing Method with Local Discontinuous Galerkin Approximation for Coupled Nonlinear Schrödinger Equations 下载免费PDF全文
This paper studies a local discontinuous Galerkin method combined with
fourth order exponential time differencing Runge-Kutta time discretization and a fourth
order conservative method for solving the nonlinear Schrödinger equations. Based on
different choices of numerical fluxes, we propose both energy-conserving and energy-dissipative
local discontinuous Galerkin methods, and have proven the error estimates
for the semi-discrete methods applied to linear Schrödinger equation. The numerical
methods are proven to be highly efficient and stable for long-range soliton computations.
Extensive numerical examples are provided to illustrate the accuracy, efficiency
and reliability of the proposed methods. 相似文献
16.
Multiscale Computations for the Maxwell–Schrödinger System in Heterogeneous Nanostructures 下载免费PDF全文
Chupeng Ma Jizu Huang Liqun Cao & Yanping Lin 《Communications In Computational Physics》2020,27(5):1443-1469
In this paper, we study the multiscale computations for the Maxwell–Schrödinger system with rapidly oscillating coefficients under the dipole approximation that describes light-matter interaction in heterogeneous nanostructures. The multiscale asymptotic method and an associated numerical algorithm for the system arepresented. We propose an alternating Crank–Nicolson finite element method for solving the homogenized Maxwell–Schödinger system and prove the existence of solutionsto the discrete system. Numerical examples are given to validate the efficiency and accuracy of the algorithm. 相似文献
17.
T. Utsumi T. Aoki J. Koga & M. Yamagiwa 《Communications In Computational Physics》2006,1(2):261-275
In this paper, we present solutions for the one-dimensional coupled nonlinear
Schrödinger (CNLS) equations by the Constrained Interpolation Profile-Basis Set
(CIP-BS) method. This method uses a simple polynomial basis set, by which physical
quantities are approximated with their values and derivatives associated with grid
points. Nonlinear operations on functions are carried out in the framework of differential
algebra. Then, by introducing scalar products and requiring the residue to be
orthogonal to the basis, the linear and nonlinear partial differential equations are reduced
to ordinary differential equations for values and spatial derivatives. The method
gives stable, less diffusive, and accurate results for the CNLS equations. 相似文献
18.
Numerical Continuation of Resonances and Bound States in Coupled Channel Schrödinger Equations 下载免费PDF全文
Przemyslaw Klosiewicz Jan Broeckhove & Wim Vanroose 《Communications In Computational Physics》2012,11(2):435-455
In this contribution, we introduce numerical continuation methods and bifurcation theory, techniques which find their roots in the study of dynamical systems, to the problem of tracing the parameter dependence of bound and resonant states of the quantum mechanical Schrödinger equation. We extend previous work on the subject [1] to systems of coupled equations. Bound and resonant states of the Schrödinger equation can be determined through the poles of the S-matrix, a quantity that can be derived from the asymptotic form of the wave function. We introduce a regularization procedure that essentially transforms the S-matrix into its inverse and improves its smoothness properties, thus making it amenable to numerical continuation. This allows us to automate the process of tracking bound and resonant states when parameters in the Schrödinger equation are varied. We have applied this approach to a number of model problems with satisfying results. 相似文献
19.
Exact Artificial Boundary Condition for the Poisson Equation in the Simulation of the 2D Schrödinger-Poisson System 下载免费PDF全文
We study the computation of ground states and time dependent solutions
of the Schrödinger-Poisson system (SPS) on a bounded domain in 2D (i.e. in two space
dimensions). On a disc-shaped domain, we derive exact artificial boundary conditions
for the Poisson potential based on truncated Fourier series expansion in θ, and propose
a second order finite difference scheme to solve the $r$-variable ODEs of the Fourier coefficients.
The Poisson potential can be solved within $\mathcal{O}$($M NlogN$) arithmetic operations
where $M,N$ are the number of grid points in $r$-direction and the Fourier bases.
Combined with the Poisson solver, a backward Euler and a semi-implicit/leap-frog
method are proposed to compute the ground state and dynamics respectively. Numerical
results are shown to confirm the accuracy and efficiency. Also we make it clear
that backward Euler sine pseudospectral (BESP) method in [33] can not be applied to
2D SPS simulation. 相似文献
20.
A Review of Transparent and Artificial Boundary Conditions Techniques for Linear and Nonlinear Schrödinger Equations 下载免费PDF全文
Xavier Antoine Anton Arnold Christophe Besse Matthias Ehrhardt & Achim Schä dle 《Communications In Computational Physics》2008,4(4):729-796
In this review article we discuss different techniques to solve numerically the time-dependent Schrödinger equation on unbounded domains. We present in detail the most recent approaches and describe briefly alternative ideas pointing out the relations between these works. We conclude with several numerical examples from different application areas to compare the presented techniques. We mainly focus on the one-dimensional problem but also touch upon the situation in two space dimensions and the cubic nonlinear case. 相似文献