Low Regularity Exponential-Type Integrators for Semilinear Schrödinger Equations

Ostermann, Alexander; Schratz, Katharina

doi:10.1007/s10208-017-9352-1

Cited by 93 publications

(140 citation statements)

References 31 publications

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“…Here D ε is defined the same as in (3.17). As used in [20], an exponential integrator is proposed by plugging (4.2) iteratively into the cubic terms (see this technique also in [66]). To describe the scheme, the following functions and operators are introduced.…”

Section: An Iterative Exponential Integrator (Iei)mentioning

confidence: 99%

“…In practice, performing all the differential operations in the above IEI method by the Fourier pseudospectral approximation with details omitted here for brevity [66], we obtain the iterative exponential integrator Fourier pseudospectral (IEI-FP) scheme. The IEI-FP is explicit, unconditionally stable, and its memory cost is O(N ) and computational cost per step is O(N ln N ).…”

Section: An Iterative Exponential Integrator (Iei)mentioning

confidence: 99%

“…The IEI-FP is explicit, unconditionally stable, and its memory cost is O(N ) and computational cost per step is O(N ln N ). As established in [66], under proper regularity of the solution u of the NKGE (2.1) [66] (which is weaker than that is needed for MTI-FP or TSF-FP), the following error bound was established for IEI-FP [66] u(·, t n ) − I N u n H 1 h m0 + τ 2 , n = 0, 1, . .…”

Section: An Iterative Exponential Integrator (Iei)mentioning

confidence: 99%

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Comparison of numerical methods for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime

Bao

Zhao

2019

Journal of Computational Physics

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Different efficient and accurate numerical methods have recently been proposed and analyzed for the nonlinear Klein-Gordon equation (NKGE) with a dimensionless parameter ε ∈ (0, 1], which is inversely proportional to the speed of light. In the nonrelativestic limit regime, i.e. 0 < ε 1, the solution of the NKGE propagates waves with wavelength at O(1) and O(ε 2 ) in space and time, respectively, which brings significantly numerical burdens in designing numerical methods. We compare systematically spatial/temporal efficiency and accuracy as well as ε-resolution (or ε-scalability) of different numerical methods including finite difference time domain methods, time-splitting method, exponential wave integrator, limit integrator, multiscale time integrator, two-scale formulation method and iterative exponential integrator. Finally, we adopt the multiscale time integrator to study the convergence rates from the NKGE to its limiting models when ε → 0 + .

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Section: An Iterative Exponential Integrator (Iei)mentioning

confidence: 99%

Section: An Iterative Exponential Integrator (Iei)mentioning

confidence: 99%

Section: An Iterative Exponential Integrator (Iei)mentioning

confidence: 99%

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Comparison of numerical methods for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime

Bao

Zhao

2019

Journal of Computational Physics

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“…The idea of twisting the variable is widely applied in the analysis of PDEs in low regularity spaces [3]. It was also widely applied in the context of numerical analysis for the Schrödinger equation [18,25], the KdV equation [15] and Klein-Gordon type equations [1,2]. For implementation issues, we impose periodic boundary conditions and refer to [11,17,23] for the corresponding well-posedness results.…”

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confidence: 99%

Two exponential-type integrators for the “good” Boussinesq equation

Ostermann

2019

Numer. Math.

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We introduce two exponential-type integrators for the "good" Bousinessq equation. They are of orders one and two, respectively, and they require lower regularity of the solution compared to the classical exponential integrators. More precisely, we will prove first-order convergence in H r for solutions in H r+1 with r > 1/2 for the derived first-order scheme. For the second integrator, we prove second-order convergence in H r for solutions in H r+3 with r > 1/2 and convergence in L 2 for solutions in H 3 . Numerical results are reported to illustrate the established error estimates. The experiments clearly demonstrate that the new exponential-type integrators are favorable over classical exponential integrators for initial data with low regularity. IntroductionConsider the "good" Boussinesq (GB) equation [4] z tt + z xxxx − z xx − (z 2 ) xx = 0, (1.1) which was originally introduced as a model for one-dimensional weakly nonlinear dispersive waves in shallow water. Similar to the well-known Korteweg-de Vries (KdV) equation and the cubic Schrödinger equation, the GB equation is one of the important models describing the interaction between nonlinearity and dispersion. The GB equation has been widely applied in many areas, e.g., plasma, coastal engineering, hydraulics studies, elastic crystals and so on.The GB equation and its various extensions have been extensively analyzed in the literature. For the well-posedness, we refer to [10-12, 17, 23, 26] and references therein. For the interaction of solitary waves, we refer to [21,22]. Many numerical methods have been developed for solving the GB equation, such as finite difference methods (FDM) [5,24], Petrov-Galerkin methods [21], mesh free methods [8], Fourier spectral methods [6,7,27] and operator splitting methods [28,29]. Regarding the numerical analysis for the GB equation,

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“…In addition, they appear in numerical analysis, for instance in the context of the modulated Fourier expansion [9,17], adiabatic integrators [17,25] as well as Lawson-type Runge-Kutta methods [24]. Recently, this technique was also established in the numerical analysis of low-regularity problems [21,28] and introduced for the highly oscillatory Klein-Gordon equation in [5]. In the latter we could develop uniformly accurate exponential-type integrators for the classical Klein-Gordon equation up to order two for the first time.…”

mentioning

confidence: 99%

Asymptotic consistent exponential-type integrators for Klein-Gordon-Schrödinger systems from relativistic to non-relativistic regimes

Baumstark¹,

Kokkala²,

Schratz³

2018

etna

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Abstract. In this paper we propose asymptotic consistent exponential-type integrators for the Klein-GordonSchrödinger system. This novel class of integrators allows us to solve the system from slowly varying relativistic up to challenging highly oscillatory non-relativistic regimes without any step size restriction. In particular, our first-and second-order exponential-type integrators are asymptotically consistent in the sense of asymptotically converging to the corresponding decoupled free Schrödinger limit system.

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Low Regularity Exponential-Type Integrators for Semilinear Schrödinger Equations

Cited by 93 publications

References 31 publications

Comparison of numerical methods for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime

Comparison of numerical methods for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime

Two exponential-type integrators for the “good” Boussinesq equation

Asymptotic consistent exponential-type integrators for Klein-Gordon-Schrödinger systems from relativistic to non-relativistic regimes

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