Optimal error estimates of fourth‐order compact finite difference methods for the nonlinear <scp>Klein–Gordon</scp> equation in the nonrelativistic regime

Zhang, Teng; Wang, Tingchun

doi:10.1002/num.22664

Cited by 5 publications

(2 citation statements)

References 32 publications

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“…High order compact finite difference methods achieve expected accuracy with less grid points, which are able to improve the spatial resolution capacity especially for

0<\varepsilon \ll 1

. The fourth‐order compact finite difference (4cFD) method is a simple scheme to attain higher spatial order with the same number of grids for the central difference method [30–32]. Recently, the 4cFD method has been used to solve the (nonlinear) Schrödinger equation [33, 34], Klein–Gorden equation [35, 36], Dirac equation [37], Burgers' equation [30] and so on.…”

Section: Introductionmentioning

confidence: 99%

Error bounds of fourth‐order compact finite difference methods for the Dirac equation in the massless and nonrelativistic regime

Feng

Ying

2022

Numerical Methods Partial

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We establish the error bounds of fourth-order compact finite difference (4cFD) methods for the Dirac equation in the massless and nonrelativistic regime, which involves a small dimensionless parameter 0 < 𝜀 ≤ 1 inversely proportional to the speed of light. In this regime, the solution propagates waves with wavelength O(𝜀) in time and O(1) in space, as well as with the wave speed O(1∕𝜀) rapid outgoing waves. We adapt the conservative and semi-implicit 4cFD methods to discretize the Dirac equation and rigorously carry out their error bounds depending explicitly on the mesh size h, time step 𝜏 and the small parameter 𝜀. Based on the error bounds, the ε-scalability of the 4cFD, which not only improves the spatial resolution capacity but also has superior accuracy than classical second-order finite difference methods. Furthermore, physical observables including the total density and current density have the same conclusions. Numerical results are provided to validate the error bounds and the dynamics of the Dirac equation with different potentials in two dimensions is presented.

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“…High order compact finite difference methods achieve expected accuracy with less grid points, which are able to improve the spatial resolution capacity especially for

0<\varepsilon \ll 1

Section: Introductionmentioning

confidence: 99%

Error bounds of fourth‐order compact finite difference methods for the Dirac equation in the massless and nonrelativistic regime

Feng

Ying

2022

Numerical Methods Partial

View full text Add to dashboard Cite

show abstract

“…High order compact finite difference methods could achieve expected accuracy with less grid points, which are able to improve the spatial resolution capacity especially for 0 < ε ≪ 1. The fourth-order compact finite difference (4cFD) method is a simple scheme to attain higher spatial order with the same number of grids for the central difference method [29,33,43]. Recently, the 4cFD method has been used to solve the (nonlinear) Schrödinger equation [24,41], Klein-Gorden equation [17,30], Dirac equation [26], Burgers' equation [29] and so on.…”

Section: Introductionmentioning

confidence: 99%

Error bounds of fourth-order compact finite difference methods for the Dirac equation in the massless and nonrelativistic regime

Feng¹,

Ying²

2021

Preprint

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We establish the error bounds of fourth-order compact finite difference (4cFD) methods for the Dirac equation in the massless and nonrelativistic regime, which involves a small dimensionless parameter 0 < ε ≤ 1 inversely proportional to the speed of light. In this regime, the solution propagates waves with wavelength O(ε) in time and O(1) in space, as well as with the wave speed O(1/ε) rapid outgoing waves. We adapt the conservative and semi-implicit 4cFD methods to discretize the Dirac equation and rigorously carry out their error bounds depending explicitly on the mesh size h, time step τ and the small parameter ε. Based on the error bounds, the ε-scalability of the 4cFD methods is h = O(ε 1/4 ) and τ = O(ε 3/2 ), which not only improves the spatial resolution capacity but also has superior accuracy than classical second-order finite difference methods. Furthermore, physical observables including the total density and current density have the same conclusions. Numerical results are provided to validate the error bounds and the dynamics of the Dirac equation with different potentials in 2D is presented.

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Uniform error bound of a conservative fourth-order compact finite difference scheme for the Zakharov system in the subsonic regime

Zhang

Wang

2022

Adv Comput Math

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Optimal error estimates of fourth‐order compact finite difference methods for the nonlinear Klein–Gordon equation in the nonrelativistic regime

Cited by 5 publications

References 32 publications

Error bounds of fourth‐order compact finite difference methods for the Dirac equation in the massless and nonrelativistic regime

Error bounds of fourth‐order compact finite difference methods for the Dirac equation in the massless and nonrelativistic regime

Error bounds of fourth-order compact finite difference methods for the Dirac equation in the massless and nonrelativistic regime

Uniform error bound of a conservative fourth-order compact finite difference scheme for the Zakharov system in the subsonic regime

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