Contents Online
Communications in Mathematical Sciences
Volume 16 (2018)
Number 5
Error estimates of finite difference time domain methods for the Klein–Gordon–Dirac system in the nonrelativistic limit regime
Pages: 1325 – 1346
DOI: https://dx.doi.org/10.4310/CMS.2018.v16.n5.a7
Authors
Abstract
In this paper, we establish error estimates of finite difference time domain (FDTD) methods for the Klein–Gordon–Dirac (KGD) system in the nonrelativistic limit regime, involving a small dimensionless parameter $0 \lt \varepsilon \ll 1$ inversely proportional to the speed of light. In this limit regime, the solution of the KGD system propagates waves with $O(\varepsilon^2)$ and $O(1)$-wavelength in time and space respectively. The high oscillation and the nonlinear coupling between the real scalar Klein–Gordon field and the complex Dirac vector field bring great challenges to the analysis of the numerical methods for the KGD system in the nonrelativistic limit regime. Four implicit/semi-implicit/explicit FDTD methods are rigorously analyzed. By applying the energy method and cut-off technique, we obtain the error bounds for the FDTD methods at $O(\tau^2 / \varepsilon^6 + h^2 / \varepsilon)$ with time step $\tau$ and mesh size $h$. Thus, in order to compute ‘correct’ solutions when $0\lt \varepsilon \ll 1$, the estimates suggest that the meshing strategy requirement of the FDTD methods is $\tau = O(\varepsilon^3)$ and $h=O(\sqrt{\varepsilon})$. In addition, numerical results are reported to support our conclusions. Our approach is valid in one, two and three dimensions.
Keywords
Klein–Gordon–Dirac system, nonrelativistic limit regime, Yukawa interaction, finite difference time domain (FDTD) methods, error estimates
2010 Mathematics Subject Classification
35Q55, 65N12, 81Q05
Received 16 February 2018
Received revised 8 June 2018
Accepted 8 June 2018
Published 19 December 2018