Discrete transparent boundary conditions for the Schrödinger equation on circular domains

Arnold, Anton; Ehrhardt, Matthias; Schulte, Maike; Sofronov, Ivan

doi:10.4310/cms.2012.v10.n3.a9

Cited by 39 publications

(36 citation statements)

References 41 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…very long time dynamics and the potential is not a confinement potential in nonlinear optics [21,23], the simple boundary condition, such as homogeneous Dirichlet or Neumann or periodic boundary condition used in the previous subsections to truncate (or approximate) the original NLSE/GPE from the whole space problem to a bounded computational domain, might bring large truncation errors except that the bounded computational domain is chosen extremely large and/or time-dependent. Thus, in order to choose a smaller computational domain which might save memory and/or computational cost, perfectly matched layers (PMLs) [60] or high-order absorbing (or artificial) boundary conditions (ABCs) [12,21,23,59,102,149,154] need to be designed and/or used at the artificial boundary so that one can truncate (or approximate) the original NLSE/GPE into a smaller bounded computational domain. Over the last 20 years, different PMLs [153,195] and/or ABCs [14,15,16,17,18,19,21,23,172,171,194] have been designed for solving the NLSE/GPE in the literatures.…”

Section: Perfectly Matched Layers And/or Absorbing Boundary Conditionsmentioning

confidence: 99%

Computational methods for the dynamics of the nonlinear Schrödinger/Gross–Pitaevskii equations

Antoine

Bao

Besse

2013

Computer Physics Communications

339

297

View full text Add to dashboard Cite

In this paper, we begin with the nonlinear Schrödinger/Gross-Pitaevskii equation (NLSE/GPE) for modeling Bose-Einstein condensation (BEC) and nonlinear optics as well as other applications, and discuss their dynamical properties ranging from time reversible, time transverse invariant, mass and energy conservation, dispersion relation to soliton solutions. Then, we review and compare different numerical methods for solving the NLSE/GPE including finite difference time domain methods and time-splitting spectral method, and discuss different absorbing boundary conditions. In addition, these numerical methods are extended to the NLSE/GPE with damping terms and/or an angular momentum rotation term as well as coupled NLSEs/GPEs. Finally, applications to simulate a quantized vortex lattice dynamics in a rotating BEC are reported.

show abstract

Section: Perfectly Matched Layers And/or Absorbing Boundary Conditionsmentioning

confidence: 99%

Computational methods for the dynamics of the nonlinear Schrödinger/Gross–Pitaevskii equations

Antoine

Bao

Besse

2013

Computer Physics Communications

339

297

View full text Add to dashboard Cite

show abstract

“…They also derived the exact artificial boundary condition for the two-dimensional Schrödinger equation by expressing the solution with Hankel's functions, implemented with a cut-off. In [21], Arnold et al proposed some discretized transparent boundary conditions for the time-dependent Schrödinger equation on a circular computational domain, and illustrated the accuracy, stability, and efficiency of the proposed method. Zhang et al [22] designed high-order artificial boundary conditions for the two-dimensional Schrödinger equation on a circular boundary by rationally approximating the kernel functions.…”

Section: Introductionmentioning

confidence: 99%

Accurate artificial boundary conditions for the semi-discretized linear Schrödinger and heat equations on rectangular domains

Yang

Pang

et al. 2018

Computer Physics Communications

View full text Add to dashboard Cite

The aim of this paper is to design some accurate artificial boundary conditions for the semi-discretized linear Schrödinger and heat equations in rectangular domains. The Laplace transform in time and discrete Fourier transform in space are applied to get the Green's functions of the semi-discretized equations in unbounded domains with single-source. An algorithm is given to compute these Green's functions accurately through some recurrence relations. Furthermore, the finite-difference method is used to discretize the reduced problem with accurate boundary conditions. Numerical simulations are presented to illustrate the accuracy of our method in the case of the linear Schrödinger and heat equations. It is shown that the reflection at the corners is correctly eliminated.

show abstract

“…This explains why simulations in the literature [10,13,31] have been restricted to some picoseconds only. We solve this problem by using a fast evaluation of the discrete convolution kernel of sum-of-exponentials, which has been presented in [6] and employed in [5] on circular domains. To our knowledge, this rather new numerical technique has not been applied to realistic device simulations so far.…”

Section: Introductionmentioning

confidence: 99%

Transient Schrödinger–Poisson simulations of a high-frequency resonant tunneling diode oscillator

Mennemann

Jüngel

Kosina

2013

Journal of Computational Physics

View full text Add to dashboard Cite

Transient simulations of a resonant tunneling diode oscillator are presented. The semiconductor model for the diode consists of a set of time-dependent Schrödinger equations coupled to the Poisson equation for the electric potential. The one-dimensional Schrödinger equations are discretized by the finite-difference Crank-Nicolson scheme using memory-type transparent boundary conditions which model the injection of electrons from the reservoirs. This scheme is unconditionally stable and reflection-free at the boundary. An efficient recursive algorithm due to Arnold, Ehrhardt, and Sofronov is used to implement the transparent boundary conditions, enabling simulations which involve a very large number of time steps. Special care has been taken to provide a discretization of the boundary data which is completely compatible with the underlying finite-difference scheme. The transient regime between two stationary states and the self-oscillatory behavior of an oscillator circuit, containing a resonant tunneling diode, is simulated for the first time.

show abstract

Discrete transparent boundary conditions for the Schrödinger equation on circular domains

Cited by 39 publications

References 41 publications

Computational methods for the dynamics of the nonlinear Schrödinger/Gross–Pitaevskii equations

Computational methods for the dynamics of the nonlinear Schrödinger/Gross–Pitaevskii equations

Accurate artificial boundary conditions for the semi-discretized linear Schrödinger and heat equations on rectangular domains

Transient Schrödinger–Poisson simulations of a high-frequency resonant tunneling diode oscillator

Contact Info

Product

Resources

About