Error estimates of a finite difference method for the Klein–Gordon–Zakharov system in the subsonic limit regime

Su, Chunmei; Yi, Wenfan

doi:10.1093/imanum/drx044

Cited by 8 publications

(7 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For simplicity of notation, we will only present the numerical method for the KGZ system on one space dimension, and the extensions to higher dimensions are straightforward. Practically, similar to most works for computation of the Zakharov-type equations [8,9,13,19,28], (2.3) is truncated on a bounded interval Ω = (a, b) with the homogeneous Dirichlet boundary condition:…”

Section: A Uniformly Accurate Finite Difference Methodsmentioning

confidence: 99%

“…To the best of our knowledge, there are few results concerning error estimates of different numerical methods for KGZ with respect to mesh size h and time step τ as well as the parameter 0 < ε ≤ 1. Recently, a conservative finite difference method (FDM) was proposed and analyzed in the subsonic limit regime [28], where it was proved that in order to obtain 'correct' oscillatory solutions, the FDM requests the meshing strategy (or ε-scalability) h = O(ε 1/2 ) and τ = O(ε 3/2 ). The reason is due to that N ε (x, t) does not converge as ε → 0 + when α = 0 or β = −1 [21,27,30] The main aim of this paper is to propose and analyze a finite difference method for KGZ, which is uniformly accurate in both space and time for 0 < ε ≤ 1.…”

Section: Introductionmentioning

confidence: 99%

“…To establish the error bounds, we apply the energy method, cut-off technique for treating the nonlinearity and the inverse estimates to bound the numerical solution, and the limiting equation via a nonlinear Klein-Gordon equation with an oscillatory potential. The error bounds of our new numerical method significantly relax the meshing strategy of the standard FDM for KGZ in the subsonic limit regime [28].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Uniform Error Bounds of a Finite Difference Method for the Zakharov System in the Subsonic Limit Regime via an Asymptotic Consistent Formulation

Bao¹,

Su²

2017

Multiscale Model. Simul.

Self Cite

View full text Add to dashboard Cite

We establish uniform error bounds of a finite difference method for the Klein-Gordon-Zakharov system (KGZ) with a dimensionless parameter ε ∈ (0, 1], which is inversely proportional to the acoustic speed. In the subsonic limit regime, i.e. 0 < ε ≪ 1, the solution propagates highly oscillatory waves in time and/or rapid outgoing initial layers in space due to the singular perturbation in the Zakharov equation and/or the incompatibility of the initial data. Specifically, the solution propagates waves with O(ε)-wavelength in time and O(1)-wavelength in space as well as outgoing initial layers in space at speed O(1/ε). This high oscillation in time and rapid outgoing waves in space of the solution cause significant burdens in designing numerical methods and establishing error estimates for KGZ. By adapting an asymptotic consistent formulation, we propose a uniformly accurate finite difference method and rigorously establish two independent error bounds at O(h 2 + τ 2 /ε) and O(h 2 + τ + ε) with h mesh size and τ time step. Thus we obtain a uniform error bound at O(h 2 + τ ) for 0 < ε ≤ 1. The main techniques in the analysis include the energy method, cut-off of the nonlinearity to bound the numerical solution, the integral approximation of the oscillatory term, and ε-dependent error bounds between the solutions of KGZ and its limiting model when ε → 0 + . Finally, numerical results are reported to confirm our error bounds.

show abstract

Section: A Uniformly Accurate Finite Difference Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Uniform Error Bounds of a Finite Difference Method for the Zakharov System in the Subsonic Limit Regime via an Asymptotic Consistent Formulation

Bao¹,

Su²

2017

Multiscale Model. Simul.

Self Cite

View full text Add to dashboard Cite

show abstract

“…To the best of our knowledge, there are few results concerning error estimates of different numerical methods for KGZ system with respect to mesh size h and time step τ as well as the parameter 0 < ε ≤ 1. Recently, a conservative finite difference method (FDM) was proposed and analyzed in the subsonic limit regime [32], where it was proved that in order to obtain "correct" oscillatory solutions, the FDM requests the meshing strategy (or ε-scalability) h = O(ε 1/2 ) and τ = O(ε 3/2 ). The reason is due to that N ε (x, t) does not converge as ε → 0 + when α = 0 or β = −1 [25,31,34] (cf.…”

Section: Introductionmentioning

confidence: 99%

Uniform error bounds of a finite difference method for the Klein-Gordon-Zakharov system in the subsonic limit regime

Bao

2017

Math. Comp.

View full text Add to dashboard Cite

We establish uniform error bounds of a finite difference method for the Klein-Gordon-Zakharov (KGZ) system with a dimensionless parameter ε ∈ (0, 1], which is inversely proportional to the acoustic speed. In the subsonic limit regime, i.e., 0 < ε 1, the solution propagates highly oscillatory waves in time and/or rapid outgoing initial layers in space due to the singular perturbation in the Zakharov equation and/or the incompatibility of the initial data. Specifically, the solution propagates waves with O(ε)-wavelength in time and O(1)-wavelength in space as well as outgoing initial layers in space at speed O(1/ε). This high oscillation in time and rapid outgoing waves in space of the solution cause significant burdens in designing numerical methods and establishing error estimates for KGZ system. By applying an asymptotic consistent formulation, we propose a uniformly accurate finite difference method and rigorously establish two independent error bounds at O(h 2 + τ 2 /ε) and O(h 2 + τ + ε) with h mesh size and τ time step. Thus we obtain a uniform error bound at O(h 2 + τ ) for 0 < ε ≤ 1. The main techniques in the analysis include the energy method, cut-off of the nonlinearity to bound the numerical solution, the integral approximation of the oscillatory term, and ε-dependent error bounds between the solutions of KGZ system and its limiting model when ε → 0 + . Finally, numerical results are reported to confirm our error bounds.

show abstract

“…This highly temporal oscillatory nature in the solution of the KGS equations (1.10) brings significant numerical difficulties, especially when 0 < ε ≪ 1 [3,28,32]. For example, classical methods may request harsh meshing strategy (or ε-scalability) in order to get 'correct' oscillatory solutions when ε ≪ 1 [8,34]. Recently, we proposed and analyzed uniform accurate finite difference methods for the Zakharov system [3] and Klein-Gordon-Zakharov system [4] in the subsonic limit regime by adopting an asymptotic consistent formulation.…”

mentioning

confidence: 99%

Uniform error estimates of a finite difference method for the Klein-Gordon-Schrödinger system in the nonrelativistic and massless limit regimes

Bao¹,

Su²

2018

Kinetic &Amp; Related Models

Self Cite

View full text Add to dashboard Cite

We establish a uniform error estimate of a finite difference method for the Klein-Gordon-Schrödinger (KGS) equations with two dimensionless parameters 0 < γ ≤ 1 and 0 < ε ≤ 1, which are the mass ratio and inversely proportional to the speed of light, respectively. In the simultaneously nonrelativistic and massless limit regimes, i.e., γ ∼ ε and ε → 0 + , the KGS equations converge singularly to the Schrödinger-Yukawa (SY) equations. When 0 < ε ≪ 1, due to the perturbation of the wave operator and/or the incompatibility of the initial data, which is described by two parameters α ≥ 0 and β ≥ −1, the solution of the KGS equations oscillates in time with O(ε)-wavelength, which requires harsh meshing strategy for classical numerical methods. We propose a uniformly accurate method based on two key points: (i) reformulating KGS system into an asymptotic consistent formulation, and (ii) applying an integral approximation of the oscillatory term. Using the energy method and the limiting equation via the SY equations with an oscillatory potential, we establish two independent error bounds at O(h 2 + τ 2 /ε) and O(h 2 + τ 2 + τ ε α * + ε 1+α * ) with h mesh size, τ time step and α * = min{1, α, 1 + β}. This implies that the method converges uniformly and optimally with quadratic convergence rate in space and uniformly in time at O(τ 4/3 ) and O(τ 1+ α * 2+α * ) for well-prepared (α * = 1) and ill-prepared (0 ≤ α * < 1) initial data, respectively. Thus the ε-scalability of the method is τ = O(1) and h = O(1) for 0 < ε ≤ 1, which is significantly better than classical methods. Numerical results are reported to confirm our error bounds. Finally, the method is applied to study the convergence rates of KGS equations to its limiting models in the simultaneously nonrelativistic and massless limit regimes.2010 Mathematics Subject Classification. Primary: 35Q55, 65M06, 65M12, 65M15.

show abstract

Error estimates of a finite difference method for the Klein–Gordon–Zakharov system in the subsonic limit regime

Cited by 8 publications

References 15 publications

Uniform Error Bounds of a Finite Difference Method for the Zakharov System in the Subsonic Limit Regime via an Asymptotic Consistent Formulation

Uniform Error Bounds of a Finite Difference Method for the Zakharov System in the Subsonic Limit Regime via an Asymptotic Consistent Formulation

Uniform error bounds of a finite difference method for the Klein-Gordon-Zakharov system in the subsonic limit regime

Uniform error estimates of a finite difference method for the Klein-Gordon-Schrödinger system in the nonrelativistic and massless limit regimes

Contact Info

Product

Resources

About