A friendly review of absorbing boundary conditions and perfectly matched layers for classical and relativistic quantum waves equations

Antoine, Xavier; Lorin, Emmanuel; Tang, Qinglin

doi:10.1080/00268976.2017.1290834

Cited by 59 publications

(58 citation statements)

References 55 publications

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“…The overall computational domain is next defined by: D = D Phy ∪ D PML . We refer to [35] for the construction of PMLs for quantum wave equations and more specifically to [38] for the derivation and analysis of PMLs for the 6 Dirac equation. Here, we outline the main features of this technique which is detailed in [42].…”

Section: Absorbing Layersmentioning

confidence: 99%

“…Finally, drastic stability conditions can lead to numerical diffusion related to the finite wave propagation speed, the speed of light c.Another numerical challenge for the Dirac equation shared by any other wave equation in real space solved on a truncated domain is the need of imposing special boundary conditions in order to avoid spurious wave reflections at the computational domain boundary. Therefore, the computational methods on truncated domains require non-reflective boundary conditions [15,34,35,36,37], absorbing or perfectly matched layers (PML) [35,38,39,40,41], or the introduction of an artificial potential [38]. On the other hand, Fourier-based methods applied on bounded domains naturally induce periodic boundary conditions, which can be problematic when dealing with delocalized wave functions.…”

mentioning

confidence: 99%

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Pseudospectral computational methods for the time-dependent Dirac equation in static curved spaces

Antoine

Fillion‐Gourdeau²,

Lorin³

et al. 2020

Journal of Computational Physics

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Pseudospectral numerical schemes for solving the Dirac equation in general static curved space are derived using a pseudodifferential representation of the Dirac equation along with a simple Fourier-basis technique. Owing to the presence of non-constant coefficients in the curved space Dirac equation, convolution products usually appear when the Fourier transform is performed. The strategy based on pseudodifferential operators allows for efficient computations of these convolution products by using an ordinary fast Fourier transform algorithm. The resulting numerical methods are efficient and have spectral convergence. Simultaneously, wave absorption at the boundary can be easily derived using absorbing layers to cope with some potential negative effects of periodic conditions inherent to spectral methods. The numerical schemes are first tested on simple systems to verify the convergence and are then applied to the dynamics of charge carriers in strained graphene. 12], tremendous efforts have been put these past two decades on the development of numerical methods for the computation of the time-dependent Dirac equation in flat Minkowski space. Real space methods such as Quantum Lattice Boltzman techniques [13,14,15,16,17], Galerkin methods [18,19,20], and pseudospectral or spectral methods [21,22,23,24,25,26,27,28] were developed to efficiently and accurately solve the Dirac equation. Semi-classical regimes were considered in [29] using Gaussian beams or Frozen Gaussian Approximations [30,31], while the non-relativistic limit has been studied in several recent papers [32,33]. The computational difficulties for solving the time-dependent Dirac equation include the fermion doubling problem, related to numerical dispersion [16], and the Zitterbewegung [1], resulting in highly oscillating solutions whose origin can be traced back to the presence of the mass term βmc 2 . Finally, drastic stability conditions can lead to numerical diffusion related to the finite wave propagation speed, the speed of light c.Another numerical challenge for the Dirac equation shared by any other wave equation in real space solved on a truncated domain is the need of imposing special boundary conditions in order to avoid spurious wave reflections at the computational domain boundary. Therefore, the computational methods on truncated domains require non-reflective boundary conditions [15,34,35,36,37], absorbing or perfectly matched layers (PML) [35,38,39,40,41], or the introduction of an artificial potential [38]. On the other hand, Fourier-based methods applied on bounded domains naturally induce periodic boundary conditions, which can be problematic when dealing with delocalized wave functions. In this respect, the technique developed in [42,43] for the Dirac equation in flat space, where a spectral method is combined with PML, is an interesting alternative. One of the goals of this article is to extend this numerical scheme to the Dirac equation in curved space.Recently, the Dirac equation in curved space-time has gained important intere...

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Section: Absorbing Layersmentioning

confidence: 99%

mentioning

confidence: 99%

Pseudospectral computational methods for the time-dependent Dirac equation in static curved spaces

Antoine

Fillion‐Gourdeau²,

Lorin³

et al. 2020

Journal of Computational Physics

Self Cite

View full text Add to dashboard Cite

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“…which is defined only in a distributional sense and using the relations above, we obtain an expression similar to (14) as follows:…”

Section: A Infinite Stripmentioning

confidence: 99%

“…13 and 14 for a comprehensive literature survey). For the approximate methods, we restrict ourselves to the pseudo-differential approach (in particular, the gauge transformation strategy 14 ) for constructing approximate nonreflecting boundary conditions referred to as the absorbing boundary conditions or artificial boundary conditions for various types of computational domains. Our goal is to understand operators of the form (∂ t − i Γ ) α , α = 1/2, −1/2, −1, .…”

Section: Introductionmentioning

confidence: 99%

On the nonreflecting boundary operators for the general two dimensional Schrödinger equation

Vaibhav

2019

Journal of Mathematical Physics

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Of the two main objectives we pursue in this paper, the first one consists in the studying operators of the form (∂ t −i Γ ) α , α = 1/2, −1/2, −1, . . . , where Γ is the Laplace-Beltrami operator. These operators arise in the context of nonreflecting boundary conditions in the pseudo-differential approach for the general Schrödinger equation. The definition of such operators is discussed in various settings and a formulation in terms of fractional operators is provided. The second objective consists in deriving corner conditions for a rectangular domain in order to make such domains amenable to the pseudo-differential approach. Stability and uniqueness of the solution is investigated for each of these novel boundary conditions.

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“…Notice that this will force us to impose absorbing conditions at the global computational domain boundary [4,11]. Then, for each subdomain Λ i,j , we will select…”

Section: Local Orbital Constructionmentioning

confidence: 99%

Domain decomposition method for the N-body time-independent and time-dependent Schrödinger equations

Lorin

2018

Numer Algor

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This paper is devoted to the derivation of a pleasingly parallel Galerkin method for the timeindependent N -body Schrödinger equation, and its time-dependent version modeling molecules subject to an external electric field [12,13,16]. In this goal, we develop a Schwarz Waveform Relaxation (SWR) Domain Decomposition Method (DDM) for the N -body Schrödinger equation. In order to optimize the efficiency and accuracy of the overall algorithm, i) we use mollifiers to regularize the singular potentials and to approximate the Schrödinger Hamiltonian, ii) we select appropriate orbitals, and iii) we carefully derive and approximate the SWR transmission conditions. Some low dimensional numerical experiments are presented to illustrate the methodology.

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A friendly review of absorbing boundary conditions and perfectly matched layers for classical and relativistic quantum waves equations

Cited by 59 publications

References 55 publications

Pseudospectral computational methods for the time-dependent Dirac equation in static curved spaces

Pseudospectral computational methods for the time-dependent Dirac equation in static curved spaces

On the nonreflecting boundary operators for the general two dimensional Schrödinger equation

Domain decomposition method for the N-body time-independent and time-dependent Schrödinger equations

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