Time‐splitting methods with charge conservation for the nonlinear Dirac equation

Li, Shu-Cun; Li, Xiang-Gui; Shi, Fang-Yuan

doi:10.1002/num.22154

Cited by 14 publications

(10 citation statements)

References 43 publications

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“…The splitting methods form an important group of methods which are quite accurate and efficient [57]. Actually, they have been widely applied for dealing with highly oscillatory systems such as the Schrödinger/nonlinear Schrödinger equations [1,8,9,22,23,55,67], the Dirac/nonlinear Dirac equations [5,6,14,54], the Maxwell-Dirac system [10,49], the Zakharov system [12,13,41,50], the Gross-Pitaevskii equation for Bose-Einstein condensation (BEC) [11], the Stokes equation [21], and the Enrenfest dynamics [32], etc.…”

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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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“…Since k 0 ≫ 1, the factors exp[∓iσ 1 k 0 ∆t] dominate the evolutions of α n and β n . Accordingly, we use the standard perturbation theory approach and seek solutions of (19) in the form:…”

Section: Unconditionally Unstable "Noise Floor"mentioning

confidence: 99%

“…Given the wide popularity of the SSM for NLS-type models, it is not surprising that it was also used extensively in studying solutions of the Gross-Neveu model (2). In numerical simulations of [17,18,19], SSM's performance for this model was favorably compared to that of other methods. In recent studies [20,21], it was shown that the SSM is capable of resolving distinctly different scales, i.e., of efficiently obtaining highly-oscillatory solutions, that occur in the non-relativistic regime of (2).…”

Section: Introductionmentioning

confidence: 99%

Study of instability of the Fourier split-step method for the massive Gross–Neveu model

Lakoba¹

2020

Journal of Computational Physics

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Stability properties of the well-known Fourier split-step method used to simulate a soliton and similar solutions of the nonlinear Dirac equations, known as the Gross-Neveu model, are studied numerically and analytically. Three distinct types of numerical instability that can occur in this case, are revealed and explained. While one of these types can be viewed as being related to the numerical instability occurring in simulations of the nonlinear Schrödinger equation, the other two types have not been studied or identified before, to the best of our knowledge. These two types of instability are unconditional, i.e. occur for arbitrarily small values of the time step. They also persist in the continuum limit, i.e. for arbitrarily fine spatial discretization. Moreover, one of them persists in the limit of an infinitely large computational domain. It is further demonstrated that similar instabilities also occur for other numerical methods applied to the Gross-Neveu soliton, as well as to certain solitons of another relativistic field theory model, the massive Thirring.

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“…The CN finite difference method and the exponential wave integrator Fourier pseudospectral method were developed by Bao et al in 2016 [16]. Besides, there are some other numerical techniques, such as multisymplectic Runge-Kutta methods [17], an efficient adaptive mesh redistribution method [18], an integrating factor method [19], and time-splitting methods with charge conservation [20]. Some experts also discussed the spin-orbitcoupled Bose-Einstein condensates [21] and the MaxwellDirac system [22][23][24] which are related to the NLD equation.…”

Section: Introductionmentioning

confidence: 99%

Compact Implicit Integration Factor Method for the Nonlinear Dirac Equation

Zhang

Shao

2017

Discrete Dynamics in Nature and Society

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A high-order accuracy numerical method is proposed to solve the (1 + 1)-dimensional nonlinear Dirac equation in this work. We construct the compact finite difference scheme for the spatial discretization and obtain a nonlinear ordinary differential system. For the temporal discretization, the implicit integration factor method is applied to deal with the nonlinear system. We therefore develop two implicit integration factor numerical schemes with full discretization, one of which can achieve fourth-order accuracy in both space and time. Numerical results are given to validate the accuracy of these schemes and to study the interaction dynamics of the nonlinear Dirac solitary waves.

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Time‐splitting methods with charge conservation for the nonlinear Dirac equation

Cited by 14 publications

References 43 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

Study of instability of the Fourier split-step method for the massive Gross–Neveu model

Compact Implicit Integration Factor Method for the Nonlinear Dirac Equation

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