Gaussian beam methods for the Dirac equation in the semi-classical regime

Huang, Zhongyi; Jin, Shi; Wu, Hao; Yin, Dongsheng

doi:10.4310/cms.2012.v10.n4.a14

Cited by 28 publications

(2 citation statements)

References 37 publications

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“…This can be circumvented by adapting the technique described in Ref. [37], by using the Gaussian beam method [63] or the frozen Gaussian approximation [64]. The generalization of these methods to curved space is now under consideration.…”

Section: Numerical Example: Gaussian Out-of-plane Deformationmentioning

confidence: 99%

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Two-dimensional Dirac matter in the semiclassical regime

Fillion‐Gourdeau¹,

Lorin²,

MacLean³

2021

Preprint

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The semiclassical regime of 2D static Dirac matter is obtained from the Dirac equation in curved space-time. To simplify the formulation, the Cartesian space-time geometry parametrization is transformed to isothermal coordinates using quasi-conformal transformations. Using this framework, it is demonstrated that to first order, the semiclassical approximation yields the relativistic Lorentz force equation with additional fictitious forces related to the space curvature and to the mass gradient. An effective graded index of refraction is defined and is directly linked to the metric component in isothermal coordinates. Semiclassical trajectories are evaluated in a simple example by solving the equation of motion and the Beltrami equation numerically.

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Section: Numerical Example: Gaussian Out-of-plane Deformationmentioning

confidence: 99%

“…A dynamical equation for S can be obtained by noticing that Eq. ( B3) is a homogeneous system of linear equations (G is a two-by-two matrix in spinor space), having a non-trivial solution only if the determinant in spinor space is zero [63]. Evaluating the determinant det G = 0 gives…”

mentioning

confidence: 99%

Two-dimensional Dirac matter in the semiclassical regime

Fillion‐Gourdeau¹,

Lorin²,

MacLean³

2021

Preprint

View full text Add to dashboard Cite

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Error bounds of fourth‐order compact finite difference methods for the Dirac equation in the massless and nonrelativistic regime

Feng

Ying

2022

Numerical Methods Partial

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We establish the error bounds of fourth-order compact finite difference (4cFD) methods for the Dirac equation in the massless and nonrelativistic regime, which involves a small dimensionless parameter 0 < 𝜀 ≤ 1 inversely proportional to the speed of light. In this regime, the solution propagates waves with wavelength O(𝜀) in time and O(1) in space, as well as with the wave speed O(1∕𝜀) rapid outgoing waves. We adapt the conservative and semi-implicit 4cFD methods to discretize the Dirac equation and rigorously carry out their error bounds depending explicitly on the mesh size h, time step 𝜏 and the small parameter 𝜀. Based on the error bounds, the ε-scalability of the 4cFD, which not only improves the spatial resolution capacity but also has superior accuracy than classical second-order finite difference methods. Furthermore, physical observables including the total density and current density have the same conclusions. Numerical results are provided to validate the error bounds and the dynamics of the Dirac equation with different potentials in two dimensions is presented.

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Error estimates of exponential wave integrators for the Dirac equation in the massless and nonrelativistic regime

Ma,

Chen

2024

Numerical Methods Partial

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We present exponential wave integrator Fourier pseudospectral (EWI‐FP) methods and establish their error estimates of the fully discrete schemes for the Dirac equation in the massless and nonrelativistic regime. This regime involves a small dimensionless parameter where , and is inversely proportional to the speed of light. The solution exhibits highly oscillatory behavior in time and rapid wave propagation in space in this regime. Specifically, the time oscillations have a wavelength of , while the spatial oscillations have a wavelength of , with a wave speed of . We employ (symmetric) exponential wave integrators for temporal derivatives and Fourier spectral discretization for spatial derivatives. We rigorously derive the error bounds which explicitly depend on the mesh size , the time step and the small dimensionless parameter . The error estimates for the EWI‐FP methods demonstrate that their meshing strategy requirement (‐scalability) necessitates setting and when . Finally, some numerical examples are provided to validate the error bounds.

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Gaussian beam methods for the Dirac equation in the semi-classical regime

Cited by 28 publications

References 37 publications

Two-dimensional Dirac matter in the semiclassical regime

Two-dimensional Dirac matter in the semiclassical regime

Error bounds of fourth‐order compact finite difference methods for the Dirac equation in the massless and nonrelativistic regime

Error estimates of exponential wave integrators for the Dirac equation in the massless and nonrelativistic regime

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