On error estimates of an exponential wave integrator sine pseudospectral method for the<scp>K</scp>lein–<scp>G</scp>ordon–<scp>Z</scp>akharov system

Zhao, Xiaofei

doi:10.1002/num.21994

Cited by 38 publications

(20 citation statements)

References 39 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For existence and uniqueness of (global) smooth solutions and physical applications of the system we refer to [7,25,26,29] and the references therein. The so-called low-plasma frequency regime c = 1 of the Klein-Gordon-Zakharov system is nowadays well understood and extensively studied numerically, see, e.g., [32] for an energy conservative finite difference method and [1,33] for exponential wave integrator methods.…”

Section: Introductionmentioning

confidence: 99%

Uniformly Accurate Oscillatory Integrators for the Klein--Gordon--Zakharov System from Low- to High-Plasma Frequency Regimes

Baumstark¹,

Schratz²

2019

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

We present a novel class of oscillatory integrators for the Klein-Gordon-Zakharov system which are uniformly accurate with respect to the plasma frequency c. Convergence holds from the slowly-varying low-plasma up to the highly oscillatory high-plasma frequency regimes without any step size restriction and, especially, uniformly in c. The introduced schemes are moreover asymptotic consistent and approximates the solutions of the corresponding Zakharov limit system in the high-plasma frequency limit (c → ∞). We in particular present the construction of the first-and second-order uniformly accurate oscillatory integrators and establish rigorous, uniform error estimates. Numerical experiments underline our theoretical convergence results.

show abstract

Section: Introductionmentioning

confidence: 99%

Uniformly Accurate Oscillatory Integrators for the Klein--Gordon--Zakharov System from Low- to High-Plasma Frequency Regimes

Baumstark¹,

Schratz²

2019

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

show abstract

“…Applying the variation-of-constants formula [34] for k ≥ 0 and s ∈ R, the general solution of (2.9) can be written as follows for any s ∈ R,…”

Section: For Any Functionmentioning

confidence: 99%

“…Evaluating (2.10) and (2.11) with s = τ and approximating the integral by the trapezoid rule or the Deuflhard-type quadrature [11,34], we immediately get…”

Section: For Any Functionmentioning

confidence: 99%

“…In more details, an unconditional full order convergence was shown for a second order operator splitting numerical scheme in the energy norm [33]: the H 2 × L 2 norm in z × z t . Nowadays, exponential time integrators have been widely applied for parabolic and hyperbolic equations [18,25,28,34]. Particularly, several efficient schemes were proposed for solving the GB equation [25] based on fourth-order exponential integrators of Runge-Kutta type.…”

mentioning

confidence: 99%

See 1 more Smart Citation

A Deuflhard-Type Exponential Integrator Fourier Pseudo-Spectral Method for the “Good” Boussinesq Equation

Yao

2020

J Sci Comput

View full text Add to dashboard Cite

We propose an exponential integrator Fourier pseudospectral method DEI-FP for solving the "Good" Boussinesq (GB) equation. The numerical scheme is based on a Deuflhardtype exponential integrator and a Fourier pseudospectral method for temporal and spatial discretizations, respectively. The scheme is fully explicit and efficient due to the fast Fourier transform. Rigorous error estimates are established for the method without any CFL-type condition constraint. In more details, the method converges quadratically and spectrally in time and space, respectively. Extensive numerical experiments are reported to confirm the theoretical analysis and to demonstrate rich dynamics of the GB equation.

show abstract

“…In the second experiment, we apply the newly developed exponential-type integrators (3.10) and (4.19) to the GB equation with rough initial data. We compare the results with the classical first-order Gautschi-type [13,14] and second-order Deuflhard-type [9,30] exponential integrators used directly to the original GB equation (1.1).…”

Section: Numerical Experimentsmentioning

confidence: 99%

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

Ostermann

2019

Numer. Math.

View full text Add to dashboard Cite

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,

show abstract

On error estimates of an exponential wave integrator sine pseudospectral method for theKlein–Gordon–Zakharov system

Cited by 38 publications

References 39 publications

Uniformly Accurate Oscillatory Integrators for the Klein--Gordon--Zakharov System from Low- to High-Plasma Frequency Regimes

Uniformly Accurate Oscillatory Integrators for the Klein--Gordon--Zakharov System from Low- to High-Plasma Frequency Regimes

A Deuflhard-Type Exponential Integrator Fourier Pseudo-Spectral Method for the “Good” Boussinesq Equation

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

Contact Info

Product

Resources

About