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

Bao, Weizhu; Dong, Xuanchun; Zhao, Xiaofei

doi:10.1137/110855004

Cited by 71 publications

(61 citation statements)

References 28 publications

(75 reference statements)

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“…In the above numerical study, they paid particular attention on the resolution of different numerical methods, i.e., meshing strategy requirement (or ε-scalability) for (1.1) when 0 < ε 1. Based on their results, in order to capture "correctly" the oscillatory solution of (1.1) in practical computations, the frequently used FDTD methods request mesh size h = O(1) and time step τ = O(ε 3 ) and the EWI-FP methods require h = O(1) and τ = O(ε 2 ), when 0 < ε 1 [5,6,8]. Thus the FDTD and EWI-FP methods converge optimally in space and time for any fixed ε = ε 0 = O(1), but they do not converge when τ = O(ε).…”

Section: Introductionmentioning

confidence: 99%

A Uniformly Accurate Multiscale Time Integrator Pseudospectral Method for the Klein--Gordon Equation in the Nonrelativistic Limit Regime

Bao¹,

Cai²,

Zhao³

2014

SIAM J. Numer. Anal.

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127

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We propose and analyze a multiscale time integrator Fourier pseudospectral (MTI-FP) method for solving the Klein-Gordon (KG) equation with a dimensionless parameter 0 < ε ≤ 1 which is inversely proportional to the speed of light. In the nonrelativistic limit regime, i.e., 0 < ε 1, the solution of the KG equation propagates waves with amplitude at O(1) and wavelength at O(ε 2 ) in time and O(1) in space, which causes significantly numerical burdens due to the high oscillation in time. The MTI-FP method is designed by adapting a multiscale decomposition by frequency (MDF) to the solution at each time step and applying an exponential wave integrator to the nonlinear Schrödinger equation with wave operator under well-prepared initial data for ε 2 -frequency and O(1)-amplitude waves and a KG-type equation with small initial data for the reminder waves in the MDF. We rigorously establish two independent error bounds in H 2 -norm to the MTI-FP method at O(h m 0 +τ 2 +ε 2 ) and O(h m 0 +τ 2 /ε 2 ) with h mesh size, τ time step, and m 0 ≥ 2 an integer depending on the regularity of the solution, which immediately imply that the MTI-FP converges uniformly and optimally in space with exponential convergence rate if the solution is smooth, and uniformly in time with linear convergence rate at O(τ ) for all ε ∈ (0, 1] and optimally with quadratic convergence rate at O(τ 2 ) in the regimes when either ε = O(1) or 0 < ε ≤ τ . Numerical results are reported to confirm the error bounds and demonstrate the efficiency and accuracy of the MTI-FP method for the KG equation, especially in the nonrelativistic limit regime.

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Section: Introductionmentioning

confidence: 99%

A Uniformly Accurate Multiscale Time Integrator Pseudospectral Method for the Klein--Gordon Equation in the Nonrelativistic Limit Regime

Bao¹,

Cai²,

Zhao³

2014

SIAM J. Numer. Anal.

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127

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“…As a benchmark for comparisons, we write down the Gautschi type exponential wave integrator sine pseudospectral method (shorted as GISP) proposed in . The scheme of the GISP is given in the three level format as: choose initial values same as before and define

ψ^{n + 1}, ϕ^{n + 1}

{\overset{ψ}{true}}^{n + 1}

{\overset{ϕ}{true}}^{n + 1}

for

n \geq 0

as (2.11), then for

n = 0

true{\begin{matrix} {\overset{ψ}{true}}_{l}^{1} = \cos true(β_{l} τ true) {\overset{ψ}{true}}_{l}^{0} + \frac{\sin true(β_{l} τ true)}{β_{l}} {\overset{true˜}{true(\overset{ψ}{true} true)}}_{l}^{0} + \frac{\cos true(β_{l} τ true) - 1}{β_{l}^{2}} {\overset{f}{true}}_{l}^{0}, \\ {\overset{ϕ}{true}}_{l}^{1} = \cos true(μ_{l} τ true) {\overset{ϕ}{true}}_{l}^{0} + \frac{\sin true(μ_{l} τ true)}{μ_{l}} {\overset{true˜}{true(\overset{ϕ}{true} true)}}_{l}^{0} + true[\cos true(μ_{l} τ true) - 1 true] {\overset{g}{true}}_{l}^{0}, \\ {\overset{true˜}{true(\overset{ψ}{true} true)}}_{l}^{1} = - β_{l} \sin true(β_{l} τ true) \overset{ψ}{true} \end{matrix}

…”

Section: An Exponential Wave Integrator Pseudospectral Methodsmentioning

confidence: 99%

“…In this section, we shall establish the rigorous error estimate results of the EWI‐SP (2.11)–(2.13) for solving the KGZ system (2.1), where the CFL‐type condition required in is removed. We shall first give the main theorem on the error bounds in the energy space

H^{1} \times L^{2}

for the variables

ψ

and

ϕ

, respectively, then prove the theorem by posterior error estimate techniques.…”

Section: Convergence Analysismentioning

confidence: 99%

“…Now, combining the Lemmas , we give the proof of Theorem by energy method with the help of mathematical induction argument , or the equivalent cutoff technique for the boundedness of numerical solutions. Proof of Theorem

n = 0

, from the scheme and assumption (A), we have

| | {\overset{e}{true}}_{ψ}^{0} | |_{L^{2}} + | | e_{ψ}^{0} | |_{H^{1}} + | | e_{ϕ}^{0} | |_{L^{2}}

≲ | | ψ^{(1)} - I_{N} ψ^{(1)} | |_{L^{2}} + | | ψ^{(0)} - I_{N} ψ^{(0)} | |_{H^{1}} + | | ϕ^{(0)} - I_{N} ϕ^{(0)} | |_{L^{2}} ≲ h^{m_{0}},

Moreover, noting , and , we get

| | e_{ρ}^{0} | |_{L^{2}} ≲ | | ϕ^{(1)} - I_{N} ϕ^{(1)} | |_{L^{2}} ≲ h^{m_{0}} .

Then by triangle inequality,

…”

Section: Convergence Analysismentioning

confidence: 99%

“…By working with the two-level scheme in the correct energy space, rigorous finite time error estimates for the proposed exponential wave integrator sine pseudospectral (EWI-SP) method are established without any CFL-type conditions. The results show that the method has the same accuracy order as the one proposed in [21], that is the second order accuracy in time and spectral accuracy in space. Numerical experiments are done to justify the theoretical results.…”

mentioning

confidence: 99%

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On error estimates of an exponential wave integrator sine pseudospectral method for theKlein–Gordon–Zakharov system

Zhao

2015

Numerical Methods Partial

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In this article, we propose an exponential wave integrator sine pseudospectral (EWI‐SP) method for solving the Klein–Gordon–Zakharov (KGZ) system. The numerical method is based on a Deuflhard‐type exponential wave integrator for temporal integrations and the sine pseudospectral method for spatial discretizations. The scheme is fully explicit, time reversible and very efficient due to the fast algorithm. Rigorous finite time error estimates are established for the EWI‐SP method in energy space with no CFL‐type conditions which show that the method has second order accuracy in time and spectral accuracy in space. Extensive numerical experiments and comparisons are done to confirm the theoretical studies. Numerical results suggest the EWI‐SP allows large time steps and mesh size in practical computing. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 266–291, 2016

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

Cited by 71 publications

References 28 publications

A Uniformly Accurate Multiscale Time Integrator Pseudospectral Method for the Klein--Gordon Equation in the Nonrelativistic Limit Regime

A Uniformly Accurate Multiscale Time Integrator Pseudospectral Method for the Klein--Gordon Equation in the Nonrelativistic Limit Regime

On error estimates of an exponential wave integrator sine pseudospectral method for theKlein–Gordon–Zakharov system

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

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