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

2020 ◽  
Author(s):  
Teng Zhang ◽  
Tingchun Wang
Keyword(s):  
Finite Difference ◽  
Error Estimates ◽  
Fourth Order ◽  
Gordon Equation ◽  
Finite Difference Methods ◽  
Optimal Error Estimates ◽  
Optimal Error ◽  
Klein Gordon ◽  
Difference Methods ◽  
Klein Gordon Equation

Optimal error estimates of finite difference time domain methods for the Klein–Gordon–Dirac system

2018 ◽  
Vol 40 (2) ◽  
pp. 1266-1293 ◽  
Author(s):  
Wenfan Yi ◽  
Yongyong Cai
Keyword(s):  
Finite Difference ◽  
Error Estimates ◽  
Time Domain ◽  
Finite Difference Time Domain ◽  
Finite Difference Methods ◽  
Optimal Error Estimates ◽  
Convergence Results ◽  
Klein Gordon ◽  
Fdtd Methods ◽  
Difference Time

Abstract We propose and analyze finite difference methods for solving the Klein–Gordon–Dirac (KGD) system. Due to the nonlinear coupling between the complex Dirac ‘wave function’ and the real Klein–Gordon field, it is a great challenge to design and analyze numerical methods for KGD. To overcome the difficulty induced by the nonlinearity, four implicit/semi-implicit/explicit finite difference time domain (FDTD) methods are presented, which are time symmetric or time reversible. By rigorous error estimates, the FDTD methods converge with second-order accuracy in both spatial and temporal discretizations, and numerical results in one dimension are reported to support our conclusion. The error analysis relies on the energy method, the special nonlinear structure in KGD and the mathematical induction. Thanks to tensor grids and discrete Sobolev inequalities, our approach and convergence results are valid in higher dimensions under minor modifications.

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