480 research outputs found
Numerical algorithms for Schrödinger equation with artificial boundary conditions
We consider a one-dimensional linear Schrödinger problem defined on an infinite domain and approximated by the Crank-Nicolson type finite difference scheme. To solve this problem numerically we restrict the computational domain by introducing the reflective, absorbing or transparent artificial boundary conditions. We investigate the conservativity of the discrete scheme with respect to the mass and energy of the solution. Results of computational experiments are presented and the efficiency of different artificial boundary conditions is discussed
Domain decomposition algorithms for the two dimensional nonlinear Schrödinger equation and simulation of Bose-Einstein condensates
International audienceIn this paper, we apply the optimized Schwarz method to the two dimensional nonlinear Schrödinger equation and extend this method to the simulation of Bose-Einstein condensates (Gross-Pitaevskii equation). We propose an extended version of the Schwartz method by introducing a preconditioned algorithm. The two algorithms are studied numerically. The experiments show that the preconditioned algorithm improves the convergence rate and reduces the computation time. In addition, the classical Robin condition and a newly constructed absorbing condition are used as transmission conditions
Formulation of a phase space exponential operator for the Wigner transport equation accounting for the spatial variation of the effective mass
A novel numerical approximation technique for the Wigner transport equation including the spatial variation of the effective mass based on the formulation of an exponential operator within the phase space is derived. In addition, a different perspective for the discretization of the phase space is provided, which finally allows flexible discretization patterns. The formalism is presented by means of a simply structured resonant tunneling diode in the stationary and transient regime utilizing a conduction band Hamilton operator. In order to account for quantum effects within heterostructure devices adequately, the corresponding spatial variation of the effective mass is considered explicitly, which is mostly disregarded in conventional methods. The results are validated by a comparison with the results obtained from the nonequilibrium Green’s function approach within the stationary regime assuming the flatband case. Additionally, the proposed approach is utilized to perform a transient analysis of the resonant tunneling diode including the self-consistent Hartree–Fock potential
All-electron time-dependent density functional theory with finite elements: Time-propagation approach
We present an all-electron method for time-dependent density functional theory which employs hierarchical nonuniform finite-element bases and the time-propagation approach. The method is capable of treating linear and nonlinear response of valence and core electrons to an external field. We also introduce (i) a preconditioner for the propagation equation, (ii) a stable way to implement absorbing boundary conditions, and (iii) a new kind of absorbing boundary condition inspired by perfectly matched layers.Peer reviewe
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Fast numerical methods for waves in periodic media
Periodic media problems widely exist in many modern application areas
like semiconductor nanostructures (e.g. quantum dots and nanocrystals),
semi-conductor superlattices, photonic crystals (PC) structures, meta
materials or Bragg gratings of surface plasmon polariton (SPP) waveguides,
etc. Often these application problems are modeled by partial differential
equations with periodic coefficients and/or periodic geometries. In order to
numerically solve these periodic structure problems efficiently one usually
confines the spatial domain to a bounded computational domain (i.e. in a
neighborhood of the region of physical interest). Hereby, the usual strategy
is to introduce so-called artificial boundaries and impose suitable boundary
conditions. For wave-like equations, the ideal boundary conditions should not
only lead to w ell-posed problems, but also mimic the perfect absorption of
waves traveling out of the computational domain through the artificial
boundaries ..
Fast numerical methods for waves in periodic media
Periodic media problems widely exist in many modern
application areas like
semiconductor nanostructures (e.g.\ quantum dots and nanocrystals),
semi-conductor superlattices,
photonic crystals (PC) structures,
meta materials or Bragg gratings of surface
plasmon polariton (SPP) waveguides, etc.
Often these application problems are modeled by partial differential
equations with periodic coefficients and/or periodic geometries.
In order to numerically solve these periodic structure problems efficiently one usually confines the spatial domain to a bounded computational domain
(i.e.\ in a neighborhood of the region of physical interest).
Hereby, the usual strategy is to introduce so-called
\emph{artificial boundaries} and impose suitable boundary conditions.
For wave-like equations, the ideal boundary conditions should not only lead to well-posed problems,
but also mimic the perfect absorption of waves traveling out of the computational domain
through the artificial boundaries.
In the first part of this chapter we present a novel analytical impedance expression
for general second order ODE problems with periodic coefficients.
This new expression for the kernel of the Dirichlet-to-Neumann mapping of the artificial boundary
conditions is then used for computing the bound states of the Schr\"odinger operator with
periodic potentials at infinity.
Other potential applications are associated with the exact artificial boundary conditions
for some time-dependent problems with periodic structures.
As an example, a two-dimensional hyperbolic equation modeling the TM polarization of
the electromagnetic field with a periodic dielectric permittivity is considered.
In the second part of this chapter we present a new numerical technique for solving periodic structure problems. This novel approach possesses several advantages.
First, it allows for a fast evaluation of the Sommerfeld-to-Sommerfeld operator for periodic
array problems. Secondly,
this computational method can also be used for bi-periodic structure problems with local defects.
In the sequel we consider several problems, such as the exterior elliptic problems with
strong coercivity, the time-dependent Schr\"odinger equation and the Helmholtz equation
with damping.
Finally, in the third part we consider
periodic arrays that are structures consisting of geometrically identical
subdomains, usually called periodic cells.
We use the Helmholtz equation as a model equation and consider
the definition and evaluation of the exact boundary mappings for general
semi-infinite arrays that are periodic in one direction for any real wavenumber.
The well-posedness of the Helmholtz equation is established via the
\emph{limiting absorption principle} (LABP).
An algorithm based on the doubling procedure of the second part of this chapter
and an extrapolation method is proposed to construct the
exact Sommerfeld-to-Sommerfeld boundary mapping.
This new algorithm benefits from its robustness and the
simplicity of implementation.
But it also suffers from the high computational cost and the
resonance wave numbers.
To overcome these shortcomings, we propose another algorithm based
on a conjecture about the asymptotic behaviour of
limiting absorption principle solutions.
The price we have to pay is the resolution of some generalized eigenvalue problem,
but still the overall computational cost is significantly reduced.
Numerical evidences show that this algorithm presents theoretically
the same results as the first algorithm.
Moreover, some quantitative comparisons between these two algorithms are given
DATA-DRIVEN MODELING AND SIMULATIONS OF SEISMIC WAVES
In recent decades, nonlocal models have been proved to be very effective in the study of complex processes and multiscale phenomena arising in many fields, such as quantum mechanics, geophysics, and cardiac electrophysiology. The fractional Laplacian(−Δ)/2 can be reviewed as nonlocal generalization of the classical Laplacian which has been widely used for the description of memory and hereditary properties of various material and process. However, the nonlocality property of fractional Laplacian introduces challenges in mathematical analysis and computation. Compared to the classical Laplacian, existing numerical methods for the fractional Laplacian still remain limited. The objectives of this research are to develop new numerical methods to solve nonlocal models with fractional Laplacian and apply them to study seismic wave modeling in both homogeneous and heterogeneous media.
To this end, we have developed two classes of methods: meshfree pseudospectral method and operator factorization methods. Compared to the current state-of-the-art methods, both of them can achieve higher accuracy with less computational complexity. The operator factorization methods provide a general framework, allowing one to achieve better accuracy with high-degree Lagrange basis functions. The meshfree pseudospectral methods based on global radial basis functions can solve both classical and fractional Laplacians in a single scheme which are the first compatible methods for these two distinct operators. Numerical experiments have demonstrated the effectiveness of our methods on various nonlocal problems. Moreover, we present an extensive study of the variable-order Laplacian operator (−Δ)(x)/2 by using meshfree methods both analytically and numerically. Finally, we apply our numerical methods to solve seismic wave modeling and study the nonlocal effects of fractional wave equation --Abstract, p. i
Stability and convergence analysis of artificial boundary conditions for the Schrödinger equation on a rectangular domain
Based on the discrete artificial boundary condition introduced in [16] for the two-dimensional free Schrödinger equation in a computational rectangular domain, we propose to analyze the stability and convergence rate of the resulting full scheme. We prove that the global scheme is L 2-stable and that the accuracy is second-order in time, confirming then the numerical results reported in [16]
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