On Fourier Time-Splitting Methods for Nonlinear Schrödinger Equations in the Semiclassical Limit

Carles, Rémi

doi:10.1137/120892416

Cited by 23 publications

(15 citation statements)

References 24 publications

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“…4.1. The RLogSE (1.12) is then solved by CNFD, LTSP and STSP on the domain Ω = [− 16,16] and Ω = [−π, π] for Case I and Case II, respectively. To quantify the numerical errors, we introduce the error function…”

Section: Accuracy Testmentioning

confidence: 99%

Regularized numerical methods for the logarithmic Schrödinger equation

Bao

Carles

et al. 2019

Numer. Math.

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We present and analyze two numerical methods for the logarithmic Schrödinger equation (LogSE) consisting of a regularized splitting method and a regularized conservative Crank-Nicolson finite difference method (CNFD). In order to avoid numerical blow-up and/or to suppress round-off error due to the logarithmic nonlinearity in the LogSE, a regularized logarithmic Schrödinger equation (RLogSE) with a small regularized parameter 0 < ε 1 is adopted to approximate the LogSE with linear convergence rate O(ε). Then we use the Lie-Trotter splitting integrator to solve the RLogSE and establish its error bound O(τ 1/2 ln(ε −1 )) with τ > 0 the time step, which implies an error bound at O(ε + τ 1/2 ln(ε −1 )) for the LogSE by the Lie-Trotter splitting method. In addition, the CNFD is also applied to discretize the RLogSE, which conserves the mass and energy in the discretized level. Numerical results are reported to confirm our error bounds and to demonstrate rich and complicated dynamics of the LogSE.

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Section: Accuracy Testmentioning

confidence: 99%

Regularized numerical methods for the logarithmic Schrödinger equation

Bao

Carles

et al. 2019

Numer. Math.

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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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“…). 19 The residue termsĨ n 2 andĨ n 4 will be estimated first. Using the properties of R n k and S e , noticing (3.45)-(3.46), we have…”

Section: Proof Of Theorem 24mentioning

confidence: 99%

“…Introduction. The splitting technique introduced by Trotter in 1959 [46] has been widely applied in analysis and numerical simulation [2,9,10,19,20], especially in computational quantum physics. In the Hamiltonian system and general ordinary differential equations (ODEs), the splitting approach has been shown to preserve the structural/geometric properties [31,47] and are superior in many applications.…”

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confidence: 99%

“…In the Hamiltonian system and general ordinary differential equations (ODEs), the splitting approach has been shown to preserve the structural/geometric properties [31,47] and are superior in many applications. Developments of splitting type methods in solving partial differential equations (PDEs) include utilization in Schrödinger/nonlinear Schrödinger equations [2,9,10,19,20,37,45], Dirac/nonlinear Dirac equations [7,8,14,36], Maxwell-Dirac system [11,32], Zakharov system [12,13,28,34,35], Stokes equation [18], and Enrenfest dynamics [25], etc.…”

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confidence: 99%

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Super-resolution of time-splitting methods for the Dirac equation in the nonrelativistic regime

2020

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We establish error bounds of the Lie-Trotter splitting (S 1 ) and Strang splitting (S 2 ) for the Dirac equation in the nonrelativistic limit regime in the absence of external magnetic potentials, with a small parameter 0 < ε ≤ 1 inversely proportional to the speed of light. In this limit regime, the solution propagates waves with O(ε 2 ) wavelength in time. Surprisingly, we find out that the splitting methods exhibit super-resolution, in the sense of breaking the resolution constraint under the Shannon's sampling theorem, i.e. the methods can capture the solutions accurately even if the time step size τ is much larger than the sampled wavelength at O(ε 2 ). S 1 shows 1/2 order convergence uniformly with respect to ε, by establishing that there are two independent error bounds τ + ε and τ + τ /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. In addition, we show S 2 is uniformly convergent with 1/2 order rate for general time step size τ and uniformly convergent with 3/2 order rate for non-resonant time step size. Finally, numerical examples are reported to validate our findings.

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On Fourier Time-Splitting Methods for Nonlinear Schrödinger Equations in the Semiclassical Limit

Cited by 23 publications

References 24 publications

Regularized numerical methods for the logarithmic Schrödinger equation

Regularized numerical methods for the logarithmic Schrödinger equation

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

Super-resolution of time-splitting methods for the Dirac equation in the nonrelativistic regime

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