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

Baumstark, Simon; Schratz, Katharina

doi:10.1137/18m1177184

Cited by 11 publications

(16 citation statements)

References 30 publications

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“…In this section we compare various numerical schemes for the solution of the KGZ system (1). Our numerical experiments confirm that in the high plasma frequency regime c 1 the ansatz (4), based on the Zakharov limit approximation, is more efficient than directly solving the KGZ system (1) with a uniformly accurate scheme such as [5]. Furthermore, the numerical experiments underline the second-order convergence rate (in time and in c −2 ) established in Corollary 3.2.…”

Section: Some Numerical Illustrationssupporting

confidence: 73%

“…• The uniformly accurate methods for the KGZ system (1) developed in [5] which allow for a global error of order τ 2 with p = 1, 2.…”

Section: Efficiencymentioning

confidence: 99%

“…We plot the corresponding error against the computation time (in seconds) of the corresponding numerical method. The reference solution is computed via the uniformly accurate method by [5] with a very small step size…”

Section: Efficiencymentioning

confidence: 99%

“…In these highly oscillatory situations the approach via the regular limit system turned out to be more effective than other tools for highly oscillatory systems, such as Gautschi type approaches (see, e.g., [1]). In strong high plasma frequency limits c → ∞ they are also far more effective than uniformly accurate oscillatory methods which have been recently invented for a number of Klein-Gordon type systems [4,5,8]. In particular, high order uniformly accurate oscillatory schemes are numerically very expensive such that the asymptotic approach of reducing the original complex system to the corresponding limit system is far more attractive from a computational point of view.…”

Section: Introductionmentioning

confidence: 99%

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Effective Numerical Simulation of the Klein–Gordon–Zakharov System in the Zakharov Limit

Baumstark

Schneider

Schratz

2020

Trends in Mathematics

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Solving the Klein-Gordon-Zakharov (KGZ) system in the high-plasma frequency regime c 1 is numerically severely challenging due to the highly oscillatory nature or the problem. To allow reliable approximations classical numerical schemes require severe step size restrictions depending on the small parameter c −2. This leads to large errors and huge computational costs. In the singular limit c → ∞ the Zakharov system appears as the regular limit system for the KGZ system. It is the purpose of this paper to use this approximation in the construction of an effective numerical scheme for the KGZ system posed on the torus in the highly oscillatory regime c 1. The idea is to filter out the highly oscillatory phases explicitly in the solution. This allows us to play back the numerical task to solving the non-oscillatory Zakharov limit system. The latter can be solved very efficiently without any step size restrictions. The numerical approximation error is then estimated by showing that solutions of the KGZ system in this singular limit can be approximated via the solutions of the Zakharov system and by proving error estimates for the numerical approximation of the Zakharov system. We close the paper with numerical experiments which show that this method is more effective than other methods in the high-plasma frequency regime c 1.

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Section: Some Numerical Illustrationssupporting

confidence: 73%

“…• The uniformly accurate methods for the KGZ system (1) developed in [5] which allow for a global error of order τ 2 with p = 1, 2.…”

Section: Efficiencymentioning

confidence: 99%

Section: Efficiencymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Effective Numerical Simulation of the Klein–Gordon–Zakharov System in the Zakharov Limit

Baumstark

Schneider

Schratz

2020

Trends in Mathematics

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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.…”

mentioning

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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Error estimates of a Fourier integrator for the cubic Schrödinger equation at low regularity

2020

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

Cited by 11 publications

References 30 publications

Effective Numerical Simulation of the Klein–Gordon–Zakharov System in the Zakharov Limit

Effective Numerical Simulation of the Klein–Gordon–Zakharov System in the Zakharov Limit

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

Error estimates of a Fourier integrator for the cubic Schrödinger equation at low regularity

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