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 < ε ≪ 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(ε), while in space, it is at O(1) with the wave speed at O(ε^− 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, and the small parameter ε. The error bounds indicate that, to obtain the “correct” numerical solution in the massless and nonrelativistic regime, i.e., 0 < ε ≪ 1, all these finite difference methods share the same ε-scalability as time step τ = O(ε^3/2) and mesh size h = O(ε^1/2). A large number of numerical results are reported to verify the error estimates.

我们提出了四种常用的有限差分方法，并建立了无质量和非相对论体系中狄拉克方程离散化的误差界限，其中涉及与光速成反比的小无量纲参数 0 < ε ≪ 1。在无质量和非相对论状态下，解在空间中表现出快速运动，并且在时间上具有高度振荡性。具体来说，传播波的波长在时间上为O ( ε )，而在空间中为O (1)，波速为O ( ε ^{− 1}）。我们采用一种跳跃式、两种半隐式和一种保守的 Crank-Nicolson 有限差分方法在一维上对 Dirac 方程进行数值离散，并严格建立明确依赖于时间步长τ、网格大小h和小参数ε。误差界限表明，为了获得在无质量和非相对论政权“正确”的数值解，即，0 < ε «1，所有这些有限差分法共享相同的ε -scalability随着时间的步骤τ = Ö（ε ^{3 / 2} ) 和网格尺寸h = O ( ε^1/2）。报告了大量数值结果以验证误差估计。