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

Feng, Yue; Xu, Zhiguo; Yin, Jingxue

doi:10.1016/j.apnum.2021.09.018

Cited by 13 publications

(4 citation statements)

References 68 publications

(19 reference statements)

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“…Recently, the numerical methods for long-time dynamics of PDEs with weak nonlinearity have received more and more attention. The long-time dynamics of the Klein-Gordon (KG) equations and Dirac equations with weak nonlinearity or small potential are thoroughly studied in the literature [7,8,18,[20][21][22][30][31][32]. For the weak nonlinear NLSW with periodic boundary condition, an exponential wave integrator Fourier pseudo-spectral method has been proposed in [24] and proved to be uniformly accurate about ε up to the time at O(1/ε 2 ).…”

Section: Introductionmentioning

confidence: 99%

Uniform Error Bounds of an Energy-Preserving Exponential Wave Integrator Fourier Pseudo-Spectral Method for the Nonlinear Schrödinger Equation with Wave Operator and Weak Nonlinearity

2024

JCM

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Recently, the numerical methods for long-time dynamics of PDEs with weak nonlinearity have received more and more attention. For the nonlinear Schrödinger equation (NLS) with wave operator (NLSW) and weak nonlinearity controlled by a small value ε ∈ (0, 1], an exponential wave integrator Fourier pseudo-spectral (EWIFP) discretization has been developed (Guo et al., 2021) and proved to be uniformly accurate about ε up to the time at O(1/ε 2 ). However, the EWIFP method is not time symmetric and can not preserve the discrete energy. As we know, the time symmetry and energy-preservation are the important structural features of the true solution and we hope that this structure can be inherited along the numerical solution. In this work, we propose a time symmetric and energy-preserving exponential wave integrator Fourier pseudo-spectral (SEPEWIFP) method for the NLSW with periodic boundary conditions. Through rigorous error analysis, we establish uniform error bounds of the numerical solution at O(h m 0 + ε 2−β τ 2 ) up to the time at O(1/ε β ) for β ∈ [0, 2], where h and τ are the mesh size and time step, respectively, and m0 depends on the regularity conditions. The tools for error analysis mainly include cut-off technique and the standard energy method. We also extend the results on error bounds, energy-preservation and time symmetry to the oscillatory NLSW with wavelength at O(ε 2 ) in time which is equivalent to the NLSW with weak nonlinearity. Numerical experiments confirm that the theoretical results in this paper are correct. Our method is novel because that to the best of our knowledge there has not been any energy-preserving exponential wave integrator method for the NLSW.

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Section: Introductionmentioning

confidence: 99%

Uniform Error Bounds of an Energy-Preserving Exponential Wave Integrator Fourier Pseudo-Spectral Method for the Nonlinear Schrödinger Equation with Wave Operator and Weak Nonlinearity

2024

JCM

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“…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%

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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“…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 Bound of an Exponential Wave Integrator for Long-Time Dynamics of the Nonlinear Schrödinger Equation with Wave Operator

Feng¹,

Yuan²

2023

EAJAM

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We establish a uniform error bound of an exponential wave integrator Fourier pseudospectral (EWI-FP) method for the long-time dynamics of the nonlinear Schrödinger equation with wave operator (NLSW), in which the strength of the nonlinearity is characterized by ǫ 2p with ǫ ∈ (0, 1] a dimensionless parameter and p ∈ + . When 0 < ǫ ≪ 1, the long-time dynamics of the problem is equivalent to that of the NLSW with(1)-nonlinearity and (ǫ)-initial data. The NLSW is numerically solved by the EWI-FP method which combines an exponential wave integrator for temporal discretization with the Fourier pseudospectral method in space. We rigorously establish the uniform H 1 -error bound of the EWI-FP method at (h m−1 + ǫ 2p−β τ 2 ) up to the time at (1/ǫ β ) with 0 ≤ β ≤ 2p, the mesh size h, time step τ and m ≥ 2 an integer depending on the regularity of the exact solution. Finally, numerical results are provided to confirm our error estimates of the EWI-FP method and show that the convergence rate is sharp.

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

Cited by 13 publications

References 68 publications

Uniform Error Bounds of an Energy-Preserving Exponential Wave Integrator Fourier Pseudo-Spectral Method for the Nonlinear Schrödinger Equation with Wave Operator and Weak Nonlinearity

Uniform Error Bounds of an Energy-Preserving Exponential Wave Integrator Fourier Pseudo-Spectral Method for the Nonlinear Schrödinger Equation with Wave Operator and Weak Nonlinearity

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 Bound of an Exponential Wave Integrator for Long-Time Dynamics of the Nonlinear Schrödinger Equation with Wave Operator

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