Conservative difference methods for the Klein–Gordon–Zakharov equations

Ellert, M.; Grønager, Michael; Константинов, Александр; Kónya, B.; Lindemann, Jonas; Livenson, Ilja; Nielsen, Johan; Niinimäki, Marko; Smirnova, O.; Wäänänen, A.

doi:10.1016/j.cam.2006.05.008

Cited by 115 publications

(35 citation statements)

References 16 publications

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“…We remark here that the boundary conditions considered here are inspired by the inherent physical nature of the system and they have been widely used in the literature for dealing with analysis and computation of the KGZ system (see, e.g., [32] and references therein).…”

Section: Methodsmentioning

confidence: 99%

“…In addition, following the standard techniques used in [9,17,20,29,31,32] for proving energy conservation of FDTD methods, it is easy to show that it conserves the energy in the discretized level.…”

Section: Finite Difference Integrator Sine Pseudospectral Methodsmentioning

confidence: 99%

“…Thus this scheme is more efficient than ECFD-SP in practical computation, especially in two and three dimensons. Finally, we also consider here an explicit finite difference sine pseudospectral (EXFD-SP) method, which is widely used in the literature when ε = O(1) and γ = O(1) [14,24,30,32],…”

Section: Finite Difference Integrator Sine Pseudospectral Methodsmentioning

confidence: 99%

“…In addition, when ε = γ, as pointed out in [25,26] that there is no null form structure for the KGZ system [23], this suggests that the KGZ system (1.1)-(1.3) may be locally ill-posed in the energy space. Along the numerical front, Wang, Chen, and Zhang [32] presented an energy conservative finite difference method and established its error estimate.…”

Section: E(t)mentioning

confidence: 99%

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An Exponential Wave Integrator Sine Pseudospectral Method for the Klein--Gordon--Zakharov System

Bao¹,

Dong²,

Zhao³

2013

SIAM J. Sci. Comput.

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Abstract. An exponential wave integrator sine pseudospectral method is presented and analyzed for discretizing the Klein-Gordon-Zakharov (KGZ) system with two dimensionless parameters 0 < ε ≤ 1 and 0 < γ ≤ 1 which are inversely proportional to the plasma frequency and the speed of sound, respectively. The main idea in the numerical method is to apply the sine pseudospectral discretization for spatial derivatives followed by using an exponential wave integrator for temporal derivatives in phase space. The method is explicit, symmetric in time, and it is of spectral accuracy in space and second-order accuracy in time for any fixed ε = ε 0 and γ = γ 0 . In the O(1)-plasma frequency and speed of sound regime, i.e., ε = O(1) and γ = O(1), we establish rigorously error estimates for the numerical method in the energy space H 1 × L 2 . We also study numerically the resolution of the method in the simultaneous high-plasma-frequency and subsonic limit regime, i.e., (ε, γ) → 0 under ε γ. In fact, in this singular limit regime, the solution of the KGZ system is highly oscillating in time, i.e., there are propagating waves with wavelength of O(ε 2 ) and O(1) in time and space, respectively. Our extensive numerical results suggest that, in order to compute "correct" solutions in the simultaneous high-plasma-frequency and subsonic limit regime, the meshing strategy (or ε-scalability) is time step τ = O(ε 2 ) and mesh size h = O(1) independent of ε. Finally, we also observe numerically that the method has the property of near conservation of the energy over long time in practical computations.

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Section: Methodsmentioning

confidence: 99%

Section: Finite Difference Integrator Sine Pseudospectral Methodsmentioning

confidence: 99%

Section: Finite Difference Integrator Sine Pseudospectral Methodsmentioning

confidence: 99%

Section: E(t)mentioning

confidence: 99%

See 2 more Smart Citations

An Exponential Wave Integrator Sine Pseudospectral Method for the Klein--Gordon--Zakharov System

Bao¹,

Dong²,

Zhao³

2013

SIAM J. Sci. Comput.

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show abstract

“…In [1,9], non-perturbative solution and global smooth solution are obtained. Two finite difference schemes for the KGZ equations were considered in [19]. As coupled equations, the KGZ equations are similar to the Zakharov equations for which many conservative schemes have been presented ( [5-6, 10-11, 17]).…”

Section: Introductionmentioning

confidence: 99%