In the numerical simulations, we take d = 1 and the electromagnetic potentials to be
V (t, x) = 1 − x 1 + x 2 , A 1 (t, x) = (x + 1) 2 1 + x 2 , x ∈ Ω ε ,(5.5)
we further take the initial **data** as [37,47] 
φ 1 (0, x) = 1 2 e −4x 2 e iS0(x) 1 + 1 + S ′ 0 (x) 2 , φ 2 (0, x) = 1 2 e −4x 2 e iS0(x) S ′ 0 (x), x ∈ Ω ε ,(5.with S 0 (x) = 1 40 (1 + cos(2πx)) , x ∈ Ω ε . (5.7)
The problem is solved numerically on the ε-dependent interval Ω ε = (−7 − T 0 /ε, 7 + T 0 /ε) with periodic boundary conditions. The 'reference' solution Φ(t, x) = (φ 1 (t, x), φ 2 (t, x)) T is obtained numerically by the TSFP method with a very fine mesh size and a very small time step, i.e., h e = 1/512 and τ e = 10 −4 . The errors are displayed at t = 1/ε. From Table 5.6, we can observe second order convergence in space and time for the CNFD method to solve the whole space problem (cf. first row in Table 5.6). The ε-resolution of the CNFD method is h = O(ε 1/2 ) and τ = O(ε 1/2 ), which is verified through the upper triangle parts of the table above the bold diagonal line. Tables 5.7 and 5.8 display the errors of the total probability e ρ (t = 1/ε) and the current density e J (t = 1/ε) by the CNFD method, which coincide with our error analysis again. The numerical results of the other three FDTD methods are similar, and we omit them for brevity.