A convergent 2D finite-difference scheme for the Dirac–Poisson system and the simulation of graphene

Brinkman, Daniel; Heitzinger, Clemens; Markowich, Peter A.

doi:10.1016/j.jcp.2013.09.052

Cited by 32 publications

(23 citation statements)

References 27 publications

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“…Particularly, for the case where d = 1, ε = 1, V (x) ≡ 0, λ 1 = −1 and λ 2 = 0 in the choice of F(Φ), the NLDE (1.1) admits explicit soliton solutions [26,40,45,52,56,60,65,66]. In the numerical aspect, many accurate and efficient numerical methods have been proposed and analyzed, such as the finite difference time domain (FDTD) methods [19,46,59], the time-splitting Fourier spectral (TSFP) methods [10,18,39,49] and the Runge-Kutta discontinuous Galerkin methods [48].…”

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confidence: 99%

Error estimates of numerical methods for the nonlinear Dirac equation in the nonrelativistic limit regime

et al. 2016

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Super-resolution of the Lie-Trotter splitting (S 1 ) and Strang splitting (S 2 ) is rigorously analyzed for the nonlinear Dirac equation without external magnetic potentials in the nonrelativistic limit regime with a small parameter 0 < ε ≤ 1 inversely proportional to the speed of light. In this limit regime, the solution highly oscillates in time with wavelength at O(ε 2 ) in time. The splitting methods surprisingly show super-resolution, in the sense of breaking the resolution constraint under the Shannon's sampling theorem, i.e. the methods can capture the solution accurately even if the time step size τ is much larger than the sampled wavelength at O(ε 2 ). Similar to the linear case, S 1 and S 2 both exhibit 1/2 order convergence uniformly with respect to ε. Moreover, if τ is non-resonant, i.e. τ is away from certain region determined by ε, S 1 would yield an improved uniform first order O(τ ) error bound, while S 2 would give improved uniform 3/2 order convergence. Numerical results are reported to confirm these rigorous results. Furthermore, we note that super-resolution is still valid for higher order splitting methods.

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confidence: 99%

Error estimates of numerical methods for the nonlinear Dirac equation in the nonrelativistic limit regime

et al. 2016

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“…For the analytical and numerical results in the classical regime, i.e, ε = 1, we refer to [11,12,15,16,20,21,42] and references therein. In the nonrelativistic/semiclassical regime, various numerical methods have been proposed and analyzed including the finite difference time domain (FDTD) methods [2,8,31], exponential wave integrator Fourier pseudospectral (EWI-FP) method [2,4], time-splitting Fourier pseudospectral (TSFP) method [4][5][6]25], Gaussian bean method [42] and so on [18,19,22]. (x)), A 1 (t, x) = sin(2x) and Φ 0 (x) = (1/(1 + sin 2 (x)), 1/(3 + cos(x))) T for different ε.…”

Section: Introductionmentioning

confidence: 99%

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

Feng¹,

Ying²

2021

Preprint

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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 methods is h = O(ε 1/4 ) and τ = O(ε 3/2 ), 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 2D is presented.

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“…A large quantity of analytical and numerical results have been devoted in this regime in literatures. For details, we refer to [1,3,4,11,12,19,21,22,24,27] and references therein. We remark here that there have been massive numerical results for the linear/nonlinear Dirac equations in different parameter regimes, such as nonrelativistic regime [5-9, 13, 14, 18], semiclassical regime [10,25,30], etc.…”

Section: Introductionmentioning

confidence: 99%

Error estimates of finite difference methods for the Dirac equation in the massless and nonrelativistic regime

Ying

Yin

2021

Preprint

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We present four frequently used finite difference methods and establish the error bounds for the discretization of the Dirac equation in the massless and nonrelativistic regime, involving a small dimensionless parameter 0 < ε ≪ 1 inversely proportional to the speed of light. In the massless and nonrelativistic regime, the solution exhibits rapid motion in space and is highly oscillatory in time. Specifically, the wavelength of the propagating waves in time is at O(ε), while in space it is at O(1) with the wave speed at O(ε −1 ). We adopt one leap-frog, two semi-implicit, and one conservative Crank-Nicolson finite difference methods to numerically discretize the Dirac equation in one dimension and establish rigorously the error estimates which depend explicitly on the time step τ , mesh size h, as well as the small parameter ε. The error bounds indicate that, to obtain the 'correct' numerical solution in the massless and nonrelativistic regime, i.e. 0 < ε ≪ 1, all these finite difference methods share the same ε-scalability as time step τ = O(ε 3/2 ) and mesh size h = O(ε 1/2 ). A large number of numerical results are reported to verify the error estimates.

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A convergent 2D finite-difference scheme for the Dirac–Poisson system and the simulation of graphene

Cited by 32 publications

References 27 publications

Error estimates of numerical methods for the nonlinear Dirac equation in the nonrelativistic limit regime

Error estimates of numerical methods for the nonlinear Dirac equation in the nonrelativistic limit regime

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

Error estimates of finite difference methods for the Dirac equation in the massless and nonrelativistic regime

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