On a splitting method for the Zakharov system

Gauckler, Ludwig J.

doi:10.1007/s00211-017-0942-2

Cited by 7 publications

(6 citation statements)

References 22 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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“…Setting ϑ = 0 in (1.1) yields the classical Zakharov system which is described by a coupled system of a Schrödinger equation for z and a wave equation for n, see [8,9,10,22,40] and references therein. The classical Zakharov system (that is ϑ = 0 in (1.1)) is numerically extensively studied, see, e.g., [3,4,11,12,23,18,19,27,31,36]. Various schemes have been proposed in case of ϑ = 0 reaching from splitting methods up to trigonometric integrators.…”

Section: Introductionmentioning

confidence: 99%

“…This idea was recently applied to the classical Zakharov system (that is ϑ = 0 in (1.1)) where a new stable class of trigonometric integrators were introduced, see [27]. This approach was recently also successfully applied in the context of splitting schemes for the Zakharov system ( [23]). In context of the quantum Zakharov system the analysis is however more involved as our aim lies in asymptotic preserving schemes which converge in the limit ϑ → 0 to solutions of the classical Zakharov system (1.3) without any step size restriction.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic preserving trigonometric integrators for the quantum Zakharov system

Baumstark

Schratz

2020

Bit Numer Math

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We present a new class of asymptotic preserving trigonometric integrators for the quantum Zakharov system. Their convergence holds in the strong quantum regime ϑ = 1 as well as in the classical regime ϑ → 0 without imposing any step size restrictions. Moreover, the new schemes are asymptotic preserving and converge to the classical Zakharov system in the limit ϑ → 0 uniformly in the time discretization parameter. Numerical experiments underline the favorable error behavior of the new schemes with first-and second-order time convergence uniformly in ϑ, first-order asymptotic convergence in ϑ and long time structure preservation properties.2010 Mathematics Subject Classification. 65N15.

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“…Liao et al [36] formulated the TS-EWI-FP method for approximations of the coupled Schrödinger-Boussinesq system. Readers can refer to the works [2,9,20,35,43,51] for more relevant references.…”

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confidence: 99%

Time splitting combined with exponential wave integrator Fourier pseudospectral method for quantum Zakharov system

Zhang¹

2022

DCDS-B

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In this paper we develop a time splitting combined with exponential wave integrator (EWI) Fourier pseudospectral (FP) method for the quantum Zakharov system (QZS), i.e. using the FP method for spatial derivatives, a time splitting technique and an EWI method for temporal derivatives in the Schrödinger-like equation and wave-type equations, respectively. The scheme is fully explicit and efficient due to fast Fourier transform. Numerical experiments for the QZS are presented to illustrate the accuracy and capability of the method, including accuracy tests, convergence of the QZS to the classical Zakharov system in the semi-classical limit, soliton-soliton collisions and pattern dynamics of the QZS in one-dimension, as well as the blow-up phenomena of QZS in two-dimension.

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On a splitting method for the Zakharov system

Cited by 7 publications

References 22 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

Asymptotic preserving trigonometric integrators for the quantum Zakharov system

Time splitting combined with exponential wave integrator Fourier pseudospectral method for quantum Zakharov system

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