共查询到20条相似文献,搜索用时 15 毫秒
1.
Construction of the Local Structure-Preserving Algorithms for the General Multi-Symplectic Hamiltonian System
下载免费PDF全文
![点击此处可从《Communications In Computational Physics》网站下载免费的PDF全文](/ch/ext_images/free.gif)
Many partial differential equations can be written as a multi-symplectic Hamiltonian system, which has three local conservation laws, namely multi-symplectic conservation law, local energy conservation law and local momentum conservation law. In this paper, we systematically give a unified framework to construct the local structure-preserving algorithms for general conservative partial differential equations starting from the multi-symplectic formulation and using the concatenating method. We construct four multi-symplectic algorithms, two local energy-preserving algorithms and two local momentum-preserving algorithms, which are independent of the boundary conditions and can be used to integrate any partial differential equations written in multi-symplectic Hamiltonian form. Among these algorithms, some have been discussed and widely used before while most are novel schemes. These algorithms are illustrated by the nonlinear Schrödinger equation and the Klein-Gordon-Schrödinger equation. Numerical experiments are conducted to show the good performance of the proposed methods. 相似文献
2.
Absorbing Boundary Conditions for Solving N-Dimensional Stationary Schrödinger Equations with Unbounded Potentials and Nonlinearities
下载免费PDF全文
![点击此处可从《Communications In Computational Physics》网站下载免费的PDF全文](/ch/ext_images/free.gif)
Pauline Klein Xavier Antoine Christophe Besse & Matthias Ehrhardt 《Communications In Computational Physics》2011,10(5):1280-1304
We propose a hierarchy of novel absorbing boundary conditions for the one-dimensional
stationary Schrödinger equation with general (linear and nonlinear) potential.
The accuracy of the new absorbing boundary conditions is investigated numerically
for the computation of energies and ground-states for linear and nonlinear
Schrödinger equations. It turns out that these absorbing boundary conditions and
their variants lead to a higher accuracy than the usual Dirichlet boundary condition.
Finally, we give the extension of these ABCs to N-dimensional stationary Schrödinger
equations. 相似文献
3.
Wenjun Cai Huai Zhang & Yushun Wang 《Communications In Computational Physics》2016,19(5):1375-1396
This paper explores the discrete singular convolution method for Hamiltonian
PDEs. The differential matrices corresponding to two delta type kernels of the
discrete singular convolution are presented analytically, which have the properties of
high-order accuracy, band-limited structure and thus can be excellent candidates for the
spatial discretizations for Hamiltonian PDEs. Taking the nonlinear Schrödinger equation
and the coupled Schrödinger equations for example, we construct two symplectic
integrators combining this kind of differential matrices and appropriate symplectic
time integrations, which both have been proved to satisfy the square conservation
laws. Comprehensive numerical experiments including comparisons with the central
finite difference method, the Fourier pseudospectral method, the wavelet collocation
method are given to show the advantages of the new type of symplectic integrators. 相似文献
4.
Fourth Order Exponential Time Differencing Method with Local Discontinuous Galerkin Approximation for Coupled Nonlinear Schrödinger Equations
下载免费PDF全文
![点击此处可从《Communications In Computational Physics》网站下载免费的PDF全文](/ch/ext_images/free.gif)
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
5.
An Energy-Preserving Wavelet Collocation Method for General Multi-Symplectic Formulations of Hamiltonian PDEs
下载免费PDF全文
![点击此处可从《Communications In Computational Physics》网站下载免费的PDF全文](/ch/ext_images/free.gif)
In this paper, we develop a novel energy-preserving wavelet collocation
method for solving general multi-symplectic formulations of Hamiltonian PDEs. Based
on the autocorrelation functions of Daubechies compactly supported scaling functions,
the wavelet collocation method is conducted for spatial discretization. The obtained
semi-discrete system is shown to be a finite-dimensional Hamiltonian system, which
has an energy conservation law. Then, the average vector field method is used for
time integration, which leads to an energy-preserving method for multi-symplectic
Hamiltonian PDEs. The proposed method is illustrated by the nonlinear Schrödinger
equation and the Camassa-Holm equation. Since differentiation matrix obtained by
the wavelet collocation method is a cyclic matrix, we can apply Fast Fourier transform
to solve equations in numerical calculation. Numerical experiments show the high
accuracy, effectiveness and conservation properties of the proposed method. 相似文献
6.
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
7.
Local Discontinuous Galerkin Methods for the 2D Simulation of Quantum Transport Phenomena on Quantum Directional Coupler
下载免费PDF全文
![点击此处可从《Communications In Computational Physics》网站下载免费的PDF全文](/ch/ext_images/free.gif)
In this paper, we present local discontinuous Galerkin methods (LDG) to
simulate an important application of the 2D stationary Schrödinger equation called
quantum transport phenomena on a typical quantum directional coupler, which frequency
change mainly reflects in $y$-direction. We present the minimal dissipation LDG
(MD-LDG) method with polynomial basis functions for the 2D stationary Schrödinger
equation which can describe quantum transport phenomena. We also give the MD-LDG
method with polynomial basis functions in $x$-direction and exponential basis
functions in $y$-direction for the 2D stationary Schrödinger equation to reduce the computational
cost. The numerical results are shown to demonstrate the accuracy and
capability of these methods. 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.
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
10.
Towards a Theoretical Background for Strong-Scattering Inversion – Direct Envelope Inversion and Gel'fand-Levitan-Marchenko Theory
下载免费PDF全文
![点击此处可从《Communications In Computational Physics》网站下载免费的PDF全文](/ch/ext_images/free.gif)
Ru-Shan Wu 《Communications In Computational Physics》2020,28(1):41-73
Strong-scattering inversion or the inverse problem for strong scattering has
different physical-mathematical foundations from the weak-scattering case. Seismic
inversion based on wave equation for strong scattering cannot be directly solved by
Newton's local optimization method which is based on weak-nonlinear assumption.
Here I try to illustrate the connection between the Schrödinger inverse scattering (inverse problem for Schrödinger equation) by GLM (Gel'fand-Levitan-Marchenko) theory and the direct envelope inversion (DEI) using reflection data. The difference between wave equation and Schrödinger equation is that the latter has a potential independent of frequency while the former has a frequency-square dependency in the
potential. I also point out that the traditional GLM equation for potential inversion can
only recover the high-wavenumber components of impedance profile. I propose to use
the Schrödinger impedance equation for direct impedance inversion and introduce a
singular impedance function which also corresponds to a singular potential for the reconstruction of impedance profile, including discontinuities and long-wavelength velocity structure. I will review the GLM theory and its application to impedance inversion including some numerical examples. Then I analyze the recently developed multiscale direct envelope inversion (MS-DEI) and its connection to the inverse Schrödinger
scattering. It is conceivable that the combination of strong-scattering inversion (inverse Schrödinger scattering) and weak-scattering inversion (local optimization based
inversion) may create some inversion methods working for a whole range of inversion
problems in geophysical exploration. 相似文献
11.
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
![点击此处可从《Communications In Computational Physics》网站下载免费的PDF全文](/ch/ext_images/free.gif)
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
12.
Numerical Solutions of Coupled Nonlinear Schrödinger Equations by Orthogonal Spline Collocation Method
下载免费PDF全文
![点击此处可从《Communications In Computational Physics》网站下载免费的PDF全文](/ch/ext_images/free.gif)
Qing-Jiang Meng Li-Ping Yin Xiao-Qing Jin & Fang-Li Qiao 《Communications In Computational Physics》2012,12(5):1392-1416
In this paper, we present the use of the orthogonal spline collocation
method for the semi-discretization scheme of the one-dimensional coupled nonlinear Schrödinger equations. This method uses the Hermite basis functions, by which
physical quantities are approximated with their values and derivatives associated with
Gaussian points. The convergence rate with order O(h4+τ2) and the stability of the
scheme are proved. Conservation properties are shown in both theory and practice.
Extensive numerical experiments are presented to validate the numerical study under
consideration. 相似文献
13.
Computing the Ground and First Excited States of the Fractional Schrödinger Equation in an Infinite Potential Well
下载免费PDF全文
![点击此处可从《Communications In Computational Physics》网站下载免费的PDF全文](/ch/ext_images/free.gif)
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
14.
Numerical Continuation of Resonances and Bound States in Coupled Channel Schrödinger Equations
下载免费PDF全文
![点击此处可从《Communications In Computational Physics》网站下载免费的PDF全文](/ch/ext_images/free.gif)
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
15.
Renato Spigler 《Communications In Computational Physics》2022,31(5):1341-1361
The quantum lattice Boltzmann (qlB) algorithm solves the 1D Dirac equations and has been used to solve approximately the classical (i.e., non-relativistic)
Schrödinger equation. We point out that the qlB method actually approximates the
hyperbolic version of the non-relativistic Schrödinger equation, whose solution is thus
obtained at the price of an additional small error. Such an error is of order of $(ω_c\tau)^{−1},$ where $ω_c:=\frac{mc^2}{h}$ is the Compton frequency, $ħ$ being the reduced Planck constant, $m$ the rest mass of the electrons, $c$ the speed of light, and $\tau$ a chosen reference time (i.e.,
1 s), and hence it vanishes in the non-relativistic limit $c → +∞.$ This asymptotic result comes from a singular perturbation process which does not require any boundary
layer and, consequently, the approximation holds uniformly, which fact is relevant in
view of numerical approximations. We also discuss this occurrence more generally, for
some classes of linear singularly perturbed partial differential equations. 相似文献
16.
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
17.
Dirichlet-to-Neumann Mapping for the Characteristic Elliptic Equations with Symmetric Periodic Coefficients
下载免费PDF全文
![点击此处可从《Communications In Computational Physics》网站下载免费的PDF全文](/ch/ext_images/free.gif)
Jingsu Kang Meirong Zhang & Chunxiong Zheng 《Communications In Computational Physics》2014,16(4):1102-1115
Based on the numerical evidences, an analytical expression of the Dirichlet-to-Neumann mapping in the form of infinite product was first conjectured for the one-dimensional characteristic Schrödinger equation with a sinusoidal potential in [Commun. Comput. Phys., 3(3): 641-658, 2008]. It was later extended for the general second-order characteristic elliptic equations with symmetric periodic coefficients in [J. Comp.
Phys., 227: 6877-6894, 2008]. In this paper, we present a proof for this Dirichlet-to-Neumann mapping. 相似文献
18.
Novel Conservative Methods for Schrödinger Equations with Variable Coefficients over Long Time
下载免费PDF全文
![点击此处可从《Communications In Computational Physics》网站下载免费的PDF全文](/ch/ext_images/free.gif)
In this paper, we propose a wavelet collocation splitting (WCS) method,
and a Fourier pseudospectral splitting (FPSS) method as comparison, for solving one-dimensional and two-dimensional Schrödinger equations with variable coefficients in
quantum mechanics. The two methods can preserve the intrinsic properties of original
problems as much as possible. The splitting technique increases the computational efficiency. Meanwhile, the error estimation and some conservative properties are investigated. It is proved to preserve the charge conservation exactly. The global energy and
momentum conservation laws can be preserved under several conditions. Numerical
experiments are conducted during long time computations to show the performances
of the proposed methods and verify the theoretical analysis. 相似文献
19.
B. Qiao C. T. Zhou X. T. He & C. H. Lai 《Communications In Computational Physics》2008,4(5):1129-1150
This paper is concerned with the pattern dynamics of the generalized nonlinear Schrödinger equations (NSEs) related with various nonlinear physical problems in plasmas. Our theoretical and numerical results show that the higher-order nonlinear effects, acting as a Hamiltonian perturbation, break down the NSE integrability and lead to chaotic behaviors. Correspondingly, coherent structures are destroyed and replaced by complex patterns. Homoclinic orbit crossings in the phase space and stochastic partition of energy in Fourier modes show typical characteristics of the stochastic motion. Our investigations show that nonlinear phenomena, such as wave turbulence and laser filamentation, are associated with the homoclinic chaos. In particular, we found that the unstable manifolds W(u)possessing the hyperbolic fixed point correspond to an initial phase θ =45◦and 225◦, and the stable manifolds W(s) correspond to θ=135◦ and 315◦. 相似文献
20.
Laurette S. Tuckerman 《Communications In Computational Physics》2015,18(5):1336-1351
Many physical processes are described by elliptic or parabolic partial differential
equations. For linear stability problems associated with such equations, the
inverse Laplacian provides a very effective preconditioner. In addition, it is also readily
available in most scientific calculations in the form of a Poisson solver or an implicit
diffusive time step. We incorporate Laplacian preconditioning into the inverse Arnoldi
method, using BiCGSTAB to solve the large linear systems. Two successful implementations
are described: spherical Couette flow described by the Navier-Stokes equations
and Bose-Einstein condensation described by the nonlinear Schrödinger equation. 相似文献