Improved Uniform Error Bounds on Time-Splitting Methods for the Long-Time Dynamics of the Dirac Equation with Small Potentials

Bao, Weizhu; Feng, Yue; Yin, Jingxue

doi:10.1137/22m146995x

Cited by 15 publications

(7 citation statements)

References 36 publications

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“…Firstly we test the temporal and spatial error of TSSPEWIFPos I and TSSPEWIFPos II separately. Tables [13][14][15][16] show the temporal errors of the methods for different 𝜀 and k with h = 1∕2 6 . Tables 17,18 show the spatial errors of the methods for different 𝜀 and h with k = 10 −4 .…”

Section: Numerical Results Of the Oscillatory Kgd Equationmentioning

confidence: 99%

“…Recently, the numerical methods for long-time dynamics of partial differential equations (PDEs) with weak nonlinearity (or small potentials) have received more and more attention. The long-time dynamics of the KG equations and Dirac equations with weak nonlinearity or small potential are thoroughly studied in the literature [11][12][13][14][22][23][24][25]. For long time analysis of other types of equations, we refer to Reference [46,47].…”

Section: Introductionmentioning

confidence: 99%

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Structure‐preserving exponential wave integrator methods and the long‐time convergence analysis for the Klein‐Gordon‐Dirac equation with the small coupling constant

Jin

2023

Numerical Methods Partial

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Recently, the numerical methods for long-time dynamics of PDEs with weak nonlinearity (or small potentials) have received more and more attention. For the Klein-Gordon-Dirac (KGD) equation with the small coupling constant 𝜀 ∈ (0, 1], we propose two time symmetric and structure-preserving exponential wave integrator Fourier pseudo-spectral (TSSPEWIFP) methods under periodic boundary conditions. The new methods are proved to preserve the energy in discrete level and in addition the first one preserves the modified discrete mass. Through rigorous error analysis, we establish uniform error bounds of the numerical solutions at O(h m 0 + 𝜀 1−𝛽 𝜏 2 ) up to the time at O(1∕𝜀 𝛽 ) for 𝛽 ∈ [0, 1] where h and 𝜏 are the mesh size and time step, respectively, and m 0 depends on the regularity conditions. Compared with the results of existing numerical analysis, our analysis has the advantage of showing the long time numerical errors for the KGD equation with the small coupling constant. The tools for error analysis mainly include cut-off technique and the standard energy method. We also extend the results on error bounds, structure-preservation and time symmetry to the oscillatory KGD equation with wavelength at O(𝜀 𝛽 ) in time which is equivalent to the KGD equation with small coupling constant. Numerical experiments confirm that the theoretical results are correct. Our methods are novel because that to the best of our knowledge there has not been

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Section: Numerical Results Of the Oscillatory Kgd Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Structure‐preserving exponential wave integrator methods and the long‐time convergence analysis for the Klein‐Gordon‐Dirac equation with the small coupling constant

Jin

2023

Numerical Methods Partial

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“…The regularity compensation oscillation (RCO) technique has been introduced to establish the improved uniform error bounds of the time-splitting method for the (nonlinear) Schrödinger equation, nonlinear Klein-Gordon equation and Dirac equation [4,5,10]. Here, for the Lawson-type exponential integrator scheme, we apply the RCO technique to deal with the last term in the RHS of (3.19) and obtain the improved uniform error bound.…”

Section: Proof For Theorem 31mentioning

confidence: 99%

“…For the time-splitting methods applied to the NKGE with power-type nonlinearity, the improved uniform error bounds were carried out by introducing the regularity compensation oscillation (RCO) technique which controls high frequency modes by the regularity of the exact solution and low frequency modes by phase cancellation and energy method. Later, the RCO technique is extended to establish the improved error bounds for the longtime dynamics of the time-splitting methods for the (nonlinear) Schrödinger equation and (nonlinear) Dirac equation [5,10]. However, the new analysis technique takes the advantage of the polynomial nonlinearity and it can not be directly extended to deal with the non-polynomial nonlinearity.…”

Section: Introductionmentioning

confidence: 99%

Uniform Error Bounds of an Exponential Wave Integrator for the Long-Time Dynamics of the Nonlinear Klein--Gordon Equation

Feng¹,

Yi²

2021

Multiscale Model. Simul.

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We establish the improved uniform error bounds on a Lawson-type exponential integrator Fourier pseudospectral (LEI-FP) method for the long-time dynamics of sine-Gordon equation where the amplitude of the initial data is O(ε) with 0 < ε ≪ 1 a dimensionless parameter up to the time at O(1/ε 2 ). The numerical scheme combines a Lawson-type exponential integrator in time with a Fourier pseudospectral method for spatial discretization, which is fully explicit and efficient in practical computation thanks to the fast Fourier transform. By separating the linear part from the sine function and employing the regularity compensation oscillation (RCO) technique which is introduced to deal with the polynomial nonlinearity by phase cancellation, we carry out the improved error bounds for the semi-discreization at O(ε 2 τ ) instead of O(τ ) according to classical error estimates and at O(h m + ε 2 τ ) for the full-discretization up to the time Tε = T /ε 2 with T > 0 fixed. This is the first work to establish the improved uniform error bound for the long-time dynamics of the NKGE with non-polynomial nonlinearity. The improved error bound is extended to an oscillatory sine-Gordon equation with O(ε 2 ) wavelength in time and O(ε −2 ) wave speed, which indicates that the temporal error is independent of ε when the time step size is chosen as O(ε 2 ). Finally, numerical examples are shown to confirm the improved error bounds and to demonstrate that they are sharp.

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“…In recent years, long-time dynamics of dispersive partial differential equations (PDEs) including the (nonlinear) Schrödinger equation, nonlinear Klein-Gordon equation and Dirac equation with weak nonlinearity or small potential are thoroughly studied in the literature [3,4,7,15,16,17]. Exponential wave integrators and time-splitting methods are widely used to solve various semilinear evolution equations and perform well in the long-time simulations [11,14,16,18,21,23].…”

Section: Introductionmentioning

confidence: 99%

Uniform error bounds of exponential wave integrator methods for the long-time dynamics of the Dirac equation with small potentials

Feng

Yin

2022

Applied Numerical Mathematics

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Improved Uniform Error Bounds on Time-Splitting Methods for the Long-Time Dynamics of the Dirac Equation with Small Potentials

Cited by 15 publications

References 36 publications

Structure‐preserving exponential wave integrator methods and the long‐time convergence analysis for the Klein‐Gordon‐Dirac equation with the small coupling constant

Structure‐preserving exponential wave integrator methods and the long‐time convergence analysis for the Klein‐Gordon‐Dirac equation with the small coupling constant

Uniform Error Bounds of an Exponential Wave Integrator for the Long-Time Dynamics of the Nonlinear Klein--Gordon Equation

Uniform error bounds of exponential wave integrator methods for the long-time dynamics of the Dirac equation with small potentials

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