Error Bounds of the Finite Difference Time Domain Methods for the Dirac Equation in the Semiclassical Regime

Ying, Ma; Yin, Jingxue

doi:10.1007/s10915-019-01063-5

Cited by 15 publications

(13 citation statements)

References 37 publications

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“…Compared with the commonly used standard finite difference methods [2,31,32], the 4cFD methods could improve the spatial resolution capacity or attain higher order accuracy in space for a given mesh size, in other words, they need less grid points while maintaining the same accuracy.…”

Section: The Main Resultsmentioning

confidence: 99%

“…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%

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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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Section: The Main Resultsmentioning

confidence: 99%

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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“…For atomic processes in relativistic heavyion collisions, a treatment in momentum space was introduced in [39]. Additionally, there have been many studies on different regimes of the Dirac equation, such as the nonrelativistic regime [3,5,6,11], and the semiclassical regime [36].…”

Section: Introductionmentioning

confidence: 99%

A fourth-order compact time-splitting method for the Dirac equation with time-dependent potentials

Yin

2021

Journal of Computational Physics

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In this paper, we present an approach to deal with the dynamics of the Dirac equation with time-dependent electromagnetic potentials using the fourth-order compact time-splitting method (S 4c ). To this purpose, the time-ordering technique for time-dependent Hamiltonians is introduced, so that the influence of the time-dependence could be limited to certain steps which are easy to treat. Actually, in the case of the Dirac equation, it turns out that only those steps involving potentials need to be amended, and the scheme remains efficient, accurate, as well as easy to implement. Numerical examples in 1D and 2D are given to validate the scheme.

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“…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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Error Bounds of the Finite Difference Time Domain Methods for the Dirac Equation in the Semiclassical Regime

Cited by 15 publications

References 37 publications

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

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

A fourth-order compact time-splitting method for the Dirac equation with time-dependent potentials

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

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