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

Bao, Weizhu; Su, Chunmei

doi:10.1090/mcom/3278

Cited by 21 publications

(12 citation statements)

References 36 publications

(50 reference statements)

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“…We remark here that, in practical computations, in order to uniformly bound the first step value v 1 ∈ X M for ε ∈ (0,1], in the above approximation (5.9), kε −β and k 2 ε −2β are replaced by sin(kε −β ) and ksin(kε −2β ), respectively [5,8]. Similar to Section 2, following the von Neumann linear stability analysis of the classical FDTD methods for the NKGE in the nonrelativistic limit regime [5,29], we can conclude the linear stability of the above FDTD methods for oscillatory NKGE (5.3) up to the fixed time s = T 0 in the following lemma.…”

Section: Fdtd Methodsmentioning

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: Fdtd Methodsmentioning

confidence: 99%

“…Then, we can conclude that there exists a solution v * such that K n (v * )=0 by applying the Brouwer fixed point theorem [2,8,28]. In other words, the CNFD (2.3) is solvable.…”

Section: Lemma 22 (Energy Conservation)mentioning

confidence: 95%

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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“…The proofs of Lemmas 2.4 and 2.5 proceed in the analogous lines as in [2,10,38,39] and we omit the details here for brevity.…”

Section: Lemma 25 (Energy Conservation)mentioning

confidence: 99%

Long time error analysis of the fourth‐order compact finite difference methods for the nonlinear Klein–Gordon equation with weak nonlinearity

Feng

2020

Numerical Methods Partial

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We present the fourth-order compact finite difference (4cFD) discretizations for the long time dynamics of the nonlinear Klein-Gordon equation (NKGE), while the nonlinearity strength is characterized by p with a constant p ∈ N + and a dimensionless parameter ∈ (0, 1]. Based on analytical results of the lifespan of the solution, rigorous error bounds of the 4cFD methods are carried out up to the time at O(−p). We pay particular attention to how error bounds depend explicitly on the mesh size h and time step as well as the small parameter ∈ (0, 1], which indicate that, in order to obtain 'correct' numerical solutions up to the time at O(−p), the-scalability (or meshing strategy requirement) of the 4cFD methods should be taken as: h = O(p/4) and = O(p/2). It has better spatial resolution capacity than the classical second order central difference methods. By a rescaling in time, it is equivalent to an oscillatory NKGE whose solution propagates waves with wavelength at O(1) in space and O(p) in time. It is straightforward to get the error bounds of the oscillatory NKGE in the fixed time. Finally, numerical results are provided to confirm our theoretical analysis.

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“…Finally, we consider the subsonic limit α → ∞ , KGZ system to formally collapses to the nonlinear Klein‐Gordon equation

false(□ + 1 false) ψ^{\infty} = false| ψ^{\infty} |^{2} ψ^{\infty},

which still enjoys a conserved energy

E^{\infty} false(t false) = false‖ ψ^{\infty} {false‖}_{H^{1}}^{2} + false‖ \partial_{t} ψ^{\infty} {false‖}_{L^{2}}^{2} - \frac{1}{2} false‖ ψ^{\infty} {false|}_{L^{4}}^{4} .

For this problem, Bao and Su proved the convergence in W 4, ∞ ⊕ W 4, ∞ by applying an asymptotic consistent formulation, where the compatibility condition

ψ_{0}^{α} \to ψ_{0}^{\infty}

is required to cover the initial layer correction with fast time variable, because the limit Equation only admits two data

ψ_{0}^{\infty}

and

ψ_{1}^{\infty}

. Here, we apply the strategy of Masmoudi and Nakanishi treating the convergence in the limit from Zakharov system to the nonlinear Schrödinger equation, to provide a very simple proof only relying on a time‐local a priori bound and energy conservation.…”

Section: Introductionmentioning

confidence: 99%

Klein‐Gordon‐Zakharov system in energy space: Blow‐up profile and subsonic limit

Shi

Wang

2019

Math Methods in App Sciences

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In this paper, we prove finite‐time blowup in energy space for the three‐dimensional Klein‐Gordon‐Zakharov (KGZ) system by modified concavity method. We obtain the blow‐up rates of solutions in local and global space, respectively. In addition, by using the energy convergence, we study the subsonic limit of the Cauchy problem for KGZ system and prove that any finite energy solution converges to the corresponding solution of Klein‐Gordon equation in energy space.

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

Cited by 21 publications

References 36 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

Long time error analysis of the fourth‐order compact finite difference methods for the nonlinear Klein–Gordon equation with weak nonlinearity

Klein‐Gordon‐Zakharov system in energy space: Blow‐up profile and subsonic limit

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