On Time-Splitting Pseudospectral Discretization for Nonlinear Klein-Gordon Equation in Nonrelativistic Limit Regime

Dong, Xuanchun; Xu, Zhiguo; Zhao, Xiaofei

doi:10.4208/cicp.280813.190214a

Cited by 29 publications

(23 citation statements)

References 37 publications

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“…The 'exact' solution is obtained numerically by the exponential-wave integrator Fourier pseudospectral method [5,19] with a very fine mesh size and a very small time step, e.g. h e = π/2 15 and τ e = 10 −5 .…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Again, the oscillatory NKGE (5.1) is time symmetric or time reversible and conserves the energy [5,19], i.e., (x,1), respectively, of the oscillatory NKGE (5.1) with d = 1, T = (0,2π) and initial data (4.1) for different 0 < ε ≤ 1 and β. We remark here that the oscillatory nature of the oscillatory NKGE (5.1) is quite different with that of the NKGE in the nonrelativistic limit regime.…”

Section: Extension To An Oscillatory Nkgementioning

confidence: 99%

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Long Time Error Analysis of Finite Difference Time Domain Methods for the Nonlinear Klein-Gordon Equation with Weak Nonlinearity

Bao¹

2019

CiCP

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We establish error bounds of the finite difference time domain (FDTD) methods for the long time dynamics of the nonlinear Klein-Gordon equation (NKGE) with a cubic nonlinearity, while the nonlinearity strength is characterized by ε 2 with 0<ε≤1 a dimensionless parameter. When 0 < ε ≪ 1, it is in the weak nonlinearity regime and the problem is equivalent to the NKGE with small initial data, while the amplitude of the initial data (and the solution) is at O(ε). Four different FDTD methods are adapted to discretize the problem and rigorous error bounds of the FDTD methods are established for the long time dynamics, i.e. error bounds are valid up to the time at O(1/ε β ) with 0 ≤ β ≤ 2, by using the energy method and the techniques of either the cut-off of the nonlinearity or the mathematical induction to bound the numerical approximate solutions. In the error bounds, we pay particular attention to how error bounds depend explicitly on the mesh size h and time step τ as well as the small parameter ε ∈ (0,1], especially in the weak nonlinearity regime when 0 < ε ≪ 1. Our error bounds indicate that, in order to get "correct" numerical solutions up to the time at O(1/ε β ), the ε-scalability (or meshing strategy) of the FDTD methods should be taken as: h =O(ε β/2 ) and τ =O(ε β/2 ). As a by-product, our results can indicate error bounds and ε-scalability of the FDTD methods for the discretization of an oscillatory NKGE which is obtained from the case of weak nonlinearity by a rescaling in time, while its solution propagates waves with wavelength at O(1) in space and O(ε β ) in time. Extensive numerical results are reported to confirm our error bounds and to demonstrate that they are sharp.

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Section: Numerical Resultsmentioning

confidence: 99%

Section: Extension To An Oscillatory Nkgementioning

confidence: 99%

Long Time Error Analysis of Finite Difference Time Domain Methods for the Nonlinear Klein-Gordon Equation with Weak Nonlinearity

Bao¹

2019

CiCP

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“…In all, the existing numerical methods either call for some time-consuming nonlinear/linear solvers, or have low accuracy order in space or time approximatio n. Recently, the exponential wave integrators which have been well-developed originally for oscillatory ODEs from molecular dynamics and are known to have many superior properties than the FD integrators as illustrated in [10,[20][21][22], coupled with trigonometric spectral methods [27] have become very popular for solving dispersive type and wave type partial differential equations such as the nonlinear Schrödinger equations and the nonlinear Klein-Gordon equation. These methods are known for offering high spatial accuracy, efficient explicit schemes without any CFL-type constraints [17,18,34] and high resolution capacity in some limit physical regimes [7,8]. The error estimates of these methods are usually done by energy method which is standard.…”

Section: Introductionmentioning

confidence: 99%

“…The error estimates of these methods are usually done by energy method which is standard. However to rigorously establish the optimal error bound without a CFL-type condition, is not a mathematically easy task especially for coupled system [9,18]. Correct energy spaces need to be used for different components in the equations.…”

Section: Introductionmentioning

confidence: 99%

An Exponential Wave Integrator Pseudospectral Method for the Symmetric Regularized-Long-Wave Equation

Zhao¹

2016

JCM

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“…Introduction. In this paper, we consider the dimensionless nonlinear Klein-Gordon (KG) equation in d-dimensions (d = 1, 2, 3) [6,5,23,25,28,13]:…”

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confidence: 99%

On comparison of asymptotic expansion techniques for nonlinear Klein-Gordon equation in the nonrelativistic limit regime

Schratz¹,

Zhao²

2020

Discrete &Amp; Continuous Dynamical Systems - B

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This work concerns the time averaging techniques for the nonlinear Klein-Gordon (KG) equation in the nonrelativistic limit regime which have recently gained a lot of attention in numerical analysis. This is due to the fact that the solution becomes highly-oscillatory in time in this regime which causes the breakdown of classical integration schemes. To overcome this numerical burden various novel numerical methods with excellent efficiency were derived in recent years. The construction of each method thereby requests essentially the averaged model of the problem. However, the averaged model of each approach is found by different kinds of asymptotic approximation techniques reaching from the modulated Fourier expansion over the multiscale expansion by frequency up to the Chapman-Enskog expansion. In this work we give a first comparison of these recently introduced asymptotic series, reviewing their approximation validity to the KG in the asymptotic limit, their smoothness assumptions as well as their geometric properties, e.g., energy conservation and long-time behaviour of the remainder.2010 Mathematics Subject Classification. 35Q40, 34C29, 34E05, 81Q05. Key words and phrases. Nonlinear Klein-Gordon equation, nonrelativistic limit regime, time averaging, higher order expansion, long-time behavior. 2841 2842 KATHARINA SCHRATZ AND XIAOFEI ZHAO The so-called nonrelativistic limit ε → 0 of the KG equation (1.1) has been extensively studied in literature from a physical and mathematical point of view. Nowadays it is well understood that the KG equation converges to a nonlinear Schrödinger (NLS) equation when ε tends to zero. In Section 2 below we present the detailed structure of the NLS limit system. For recent analytic approximations results we refer to [36,33,34], and in the context of Birkhoff normal form transformations in particular to the recent work [38], as well as the references therein. From a numerical point of view, however, the Klein-Gordon equation in the nonrelativistic was an open problem for a long time. Classical methods are not able to resolve the highly-oscillatory nature of the solution which leads to severe step size restrictions (at least at order τ = O(ε −2 )) and huge computational costs ([6]). This failure of classical integration schemes triggered intensive studies on the numerical averaging of the model, serving to describe better asymptotic behaviour of the solution and to design efficient numerical approximations. Based on this idea, very recently novel numerical integrators were proposed which allow us to numerically solve the KG equation (1.1) in the non relativistic regime ε → 0, capturing the highly-oscillatory behaviour of the solution.Three novel schemes for the Klein-Gordon equation in the non relativistic limit regime were presented so far in literature [28,5,18]. Each of the proposed numerical schemes for KG is essentially based on an asymptotic expansion techinque for the averaged model, and each of the asymptotic expansion is mathematically obtained by different analytic techn...

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On Time-Splitting Pseudospectral Discretization for Nonlinear Klein-Gordon Equation in Nonrelativistic Limit Regime

Cited by 29 publications

References 37 publications

Long Time Error Analysis of Finite Difference Time Domain Methods for the Nonlinear Klein-Gordon Equation with Weak Nonlinearity

Long Time Error Analysis of Finite Difference Time Domain Methods for the Nonlinear Klein-Gordon Equation with Weak Nonlinearity

An Exponential Wave Integrator Pseudospectral Method for the Symmetric Regularized-Long-Wave Equation

On comparison of asymptotic expansion techniques for nonlinear Klein-Gordon equation in the nonrelativistic limit regime

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