无质量非相对论性Dirac方程有限差分法的误差估计
Error estimates of finite difference methods for the Dirac equation in the massless and nonrelativistic regime
摘要：
我们提出了四种常用的有限差分法，并建立了误差界的Dirac方程的离散化在无质量和非相对论制度，涉及一个小的无量纲参数$0< \varepsilon \ll 1$成反比的光速。在无质量和非相对论的制度，解决方案在空间中表现出快速运动，并在时间上是高度振荡。具体地说，传播波的波长在时间上是$O(\varepsilon)$，而在空间上是$O(1)$，波速是$O(\varepsilon^{-1}).$。我们采用一种蛙跳、两种半隐式和一种保守的Crank-Nicolson有限差分法对一维Dirac方程进行数值离散，并建立了严格的误差估计，它显式地依赖于时间步长$τ$，网格尺寸$h$，以及小参数$\varepsilon$。误差界表明，要获得“正确的”数值解的无质量和非相对论制度，即$0<\varepsilon\ll1$，所有这些有限差分法共享相同的$\varepsilon$ -可扩展性的时间步长$τ=O(\varepsilon^{3/2})$和网格大小$h=O(\varepsilon^{1/2})$。大量的数值结果报告，以验证误差估计。
英文摘要：
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 $τ$, 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 $τ=O(\varepsilon^{3/2})$ and mesh size $h=O(\varepsilon^{1/2})$. A large number of numerical results are reported to verify the error estimates.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
