We present four frequently used finite difference methods and establish the error bounds for the discretization of the Dirac equation in the massless and nonrelativistic regime, involving a small dimensionless parameter $0< \varepsilon \ll 1$ inversely proportional to the speed of light. In the massless and nonrelativistic regime, the solution exhibits rapid motion in space and is highly oscillatory in time. Specifically, the wavelength of the propagating waves in time is at $O(\varepsilon)$, while in space it is at $O(1)$ with the wave speed at $O(\varepsilon^{-1}).$ We adopt one leap-frog, two semi-implicit, and one conservative Crank-Nicolson finite difference methods to numerically discretize the Dirac equation in one dimension and establish rigorously the error estimates which depend explicitly on the time step $\tau$, mesh size $h$, as well as the small parameter $\varepsilon$. The error bounds indicate that, to obtain the `correct' numerical solution in the massless and nonrelativistic regime, i.e. $0<\varepsilon\ll1$, all these finite difference methods share the same $\varepsilon$-scalability as time step $\tau=O(\varepsilon^{3/2})$ and mesh size $h=O(\varepsilon^{1/2})$. A large number of numerical results are reported to verify the error estimates.
翻译:在无质量和非相对化制度中,我们经常提出四种常用的有限差异方法,并为Dirac方程式的离散设定错误界限,涉及一个小维度参数 $0 < varepsilon =ll 1美元,与光速成反比。在无质量和非相对化制度中,解决方案在空间中表现出快速移动,且在时间上高度混杂。具体地说,在无质量和非相对化制度中,传播波的波长为O(varepsilon)美元,而在空间中,其波速为1美元(1美元),以O(vareprepsilon) =-1美元。我们采用了一个跳法,两个半不透明,以及一个保守的crank-Nicolson 差异方法,将Dirac方程式从一个方面进行数字离散,并严格地确定错误估计,这明确取决于时间步的美元, meshech = $美元,而在小参数 $\\ vareplon $。错误界限表明,要获得Orevilal1 ral ral ral ral ral =x rus rus ral ral ral =x ral =x =x。