7 results on '"Zheng, Chunxiong"'
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2. Fast and stable evaluation of the exact absorbing boundary condition for the semi-discrete linear Schrödinger equation in unbounded domains.
- Author
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Hu, Jiashun and Zheng, Chunxiong
- Subjects
- *
SCHRODINGER equation , *BOUNDARY value problems , *MATHEMATICAL domains , *NUMERICAL analysis , *LINEAR equations , *MATHEMATICAL transformations - Abstract
This paper is concerned with the numerical solution of the one-dimensional semi-discrete linear Schrödinger equation in unbounded domains. In order to compute the solution on the domain of physical interest, the artificial boundary method is applied to transform the original unbounded domain problem into an initial boundary value problem on a truncated finite domain. We prove the stability of the truncated semi-discrete problem. Then, a fast algorithm is proposed to approximate the nonlocal absorbing boundary condition. The novelty of this fast algorithm is that the stability of the approximate truncated semi-discrete problem is automatically maintained. In the end, numerical examples are presented to demonstrate the performance of the proposed algorithm. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
3. Fast numerical methods for waves in periodic media
- Author
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Ehrhardt, Matthias and Zheng, Chunxiong
- Subjects
hyperbolic equation ,artificial boundary conditions ,band structure ,Floquet-Bloch theory ,65M99 ,high-order finite elements ,Schrödinger equation ,35B27 ,81-08 ,Dirichlet-to-Neumann maps ,35J05 ,35Q60 ,Robin-to-Robin maps ,Helmholtz equation ,periodic potential ,unbounded domain - Abstract
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 emphartificial 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 Schrödinger 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 Schrödinger 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 emphlimiting 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.
- Published
- 2009
- Full Text
- View/download PDF
4. Implementing exact absorbing boundary condition for the linear one-dimensional Schrödinger problem with variable potential by Titchmarsh--Weyl theory
- Author
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Ehrhardt, Matthias and Zheng, Chunxiong
- Subjects
65M99 ,unbound domain ,Schrödinger equation ,81-08 ,variable potential ,absorbing boundary conditions ,unbounded domain ,Titchmarsh-Weyl m-function - Abstract
A new approach for simulating the solution of the time-dependent Schrödinger equation with a general variable potential will be proposed. The key idea is to approximate the Titchmarsh-Weyl m-function (exact Dirichlet-to-Neumann operator) by a rational function with respect to a suitable spectral parameter. With the proposed method we can overcome the usual high-frequency restriction for absorbing boundary conditions of general variable potential problems. We end up with a fast computational algorithm for absorbing boundary conditions that are accurate for the full frequency band.
- Published
- 2009
- Full Text
- View/download PDF
5. Numerical simulation of waves in periodic structures
- Author
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Ehrhardt, Matthias, Han, Houde, and Zheng, Chunxiong
- Subjects
85.35.-p ,85.35.Be ,band structure ,Floquet-Bloch theory ,65M99 ,high-order finite elements ,Schrödinger equation ,35B27 ,02.70.Bf ,42.82.Et ,31.15.-p ,Dirichlet-to-Neumann maps ,35J05 ,35Q60 ,Dirichletto-Neumann maps ,Robin-to-Robin maps ,Helmholtz equation ,periodic media - Abstract
In this work we present a new numerical technique for solving periodic structure problems. This new approach possesses several advantages. First, it allows for a fast evaluation of the Robin-to-Robin operator for periodic array problems. Secondly, this computational method can also be used for bi-periodic structure problems with local defects. Our strategy is an improvement of the recently developed recursive doubling process by Yuan and Lu. In this paper we consider several problems, such as the exterior elliptic problems with strong coercivity, the time-dependent Schrödinger equation and finally the Helmholtz equation with damping.
- Published
- 2008
- Full Text
- View/download PDF
6. Exact artificial boundary conditions for problems with periodic structures
- Author
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Ehrhardt, Matthias and Zheng, Chunxiong
- Subjects
- *
BOUNDARY value problems , *DIFFERENTIAL equations , *ELECTROMAGNETIC fields , *SCHRODINGER equation - Abstract
Abstract: Based on the work of Zheng on the artificial boundary condition for the Schrödinger equation with sinusoidal potentials at infinity, an analytical impedance expression is presented for general second-order ODE problems with periodic coefficients and its validity is shown to be strongly supported by numerical evidences. 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 Schrödinger 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. [Copyright &y& Elsevier]
- Published
- 2008
- Full Text
- View/download PDF
7. Gaussian beam formulations and interface conditions for the one-dimensional linear Schrödinger equation
- Author
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Yin, Dongsheng and Zheng, Chunxiong
- Subjects
- *
GAUSSIAN beams , *SCHRODINGER equation , *WAVE equation , *NUMERICAL analysis , *THEORY of wave motion , *FUNCTIONAL analysis - Abstract
Abstract: Gaussian beams are asymptotic solutions of linear wave-like equations in the high frequency regime. This paper is concerned with the beam formulations for the Schrödinger equation and the interface conditions while beams pass through a singular point of the potential function. The equations satisfied by Gaussian beams up to the fourth order are given explicitly. When a Gaussian beam arrives at a singular point of the potential, it typically splits into a reflected wave and a transmitted wave. Under suitable conditions, the reflected wave and/or the transmitted wave will maintain a beam profile. We study the interface conditions which specify the relations between the split waves and the incident Gaussian beam. Numerical tests are presented to validate the beam formulations and interface conditions. [Copyright &y& Elsevier]
- Published
- 2011
- Full Text
- View/download PDF
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