To construct a fully discrete scheme, we divide the time interval [0, T] into \(N_t+2\) uniform intervals, i.e., we have discrete times \(0=t_0<t_1<\cdots < t_{N_t+2}=T.\)
Approximating those time directives properly in the semi-discrete schemes (8), (10), (12), (14), (15), and (16), we can obtain the following fully-discrete scheme: Given initial approximations \(E_{x,i+\frac{1}{2},j}^{0},E_{y,i,j+\frac{1}{2}}^{0}\), \(H_{i+\frac{1}{2},j+\frac{1}{2}}^{\frac{1}{2}}, J_{x,i+\frac{1}{2},j}^{\frac{1}{2}},J_{y,i,j+\frac{1}{2}}^{\frac{1}{2}}\), \(K_{i+\frac{1}{2},j+\frac{1}{2}}^{1},\) for any \(0\le n\le N_t\), solve \(E_{x,i+\frac{1}{2},j}^{n+1},E_{y,i,j+\frac{1}{2}}^{n+1}\), \(H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{3}{2}}, J_{x,i+\frac{1}{2},j}^{n+\frac{3}{2}},J_{y,i,j+\frac{1}{2}}^{n+\frac{3}{2}}\), \(K_{i+\frac{1}{2},j+\frac{1}{2}}^{n+2}\) from:
$$\begin{aligned}&\epsilon _0\frac{E_{x,i+\frac{1}{2},j}^{n+1}-E_{x,i+\frac{1}{2},j}^{n}}{\tau } =\frac{H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}} -H_{i+\frac{1}{2},j-\frac{1}{2}}^{n+\frac{1}{2}}}{y_{j+\frac{1}{2}}-y_{j-\frac{1}{2}}} -J_{x,i+\frac{1}{2},j}^{n+\frac{1}{2}}, \end{aligned}$$
(34)
$$\begin{aligned}&\epsilon _0\frac{E_{y,i,j+\frac{1}{2}}^{n+1}-E_{y,i,j+\frac{1}{2}}^{n}}{\tau } =-\frac{H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}} -H_{i-\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}}}{x_{i+\frac{1}{2}}-x_{i-\frac{1}{2}}} -J_{y,i,j+\frac{1}{2}}^{n+\frac{1}{2}}, \end{aligned}$$
(35)
$$\begin{aligned}&\mu _0\frac{H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{3}{2}} -H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}}}{\tau } =-\frac{E_{y,i+1,j+\frac{1}{2}}^{n+1}-E_{y,i,j+\frac{1}{2}}^{n+1}}{x_{i+1}-x_i}\nonumber \\&\quad +\frac{E_{x,i+\frac{1}{2},j+1}^{n+1}-E_{x,i+\frac{1}{2},j}^{n+1}}{y_{j+1}-y_j} -K_{i+\frac{1}{2},j+\frac{1}{2}}^{n+1}, \end{aligned}$$
(36)
$$\begin{aligned}&\frac{1}{\epsilon _0\omega _{pe}^2}\frac{J_{x,i+\frac{1}{2},j}^{n+\frac{3}{2}} -J_{x,i+\frac{1}{2},j}^{n+\frac{1}{2}}}{\tau } + \frac{\Gamma _e}{\epsilon _0\omega _{pe}^2}\frac{J_{x,i+\frac{1}{2},j}^{n+\frac{3}{2}} +J_{x,i+\frac{1}{2},j}^{n+\frac{1}{2}}}{2} =E_{x,i+\frac{1}{2},j}^{n+1}, \end{aligned}$$
(37)
$$\begin{aligned}&\frac{1}{\epsilon _0\omega _{pe}^2}\frac{J_{y,i,j+\frac{1}{2}}^{n+\frac{3}{2}} -J_{y,i,j+\frac{1}{2}}^{n+\frac{1}{2}}}{\tau } + \frac{\Gamma _e}{\epsilon _0\omega _{pe}^2}\frac{J_{y,i,j+\frac{1}{2}}^{n+\frac{3}{2}} +J_{y,i,j+\frac{1}{2}}^{n+\frac{1}{2}}}{2} =E_{y,i,j+\frac{1}{2}}^{n+1}, \end{aligned}$$
(38)
$$\begin{aligned}&\frac{1}{\mu _0\omega _{pm}^2} \frac{K_{i+\frac{1}{2},j+\frac{1}{2}}^{n+2}-K_{i+\frac{1}{2},j+\frac{1}{2}}^{n+1}}{\tau } + \frac{\Gamma _m}{\mu _0\omega _{pm}^2}\frac{K_{i+\frac{1}{2},j+\frac{1}{2}}^{n+2} +K_{i+\frac{1}{2},j+\frac{1}{2}}^{n+1}}{2} =H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{3}{2}}. \nonumber \\ \end{aligned}$$
(39)
Let \(C_v=1/\sqrt{\epsilon _0\mu _0}\) be the wave propagation speed in free space. For any grid function \(u_{i,j}\), let us denote the backward difference operators \(\nabla _x\) and \(\nabla _y\):
$$\begin{aligned} \nabla _x u_{i+1,j}=\frac{u_{i+1,j}-u_{i,j}}{x_{i+1}-x_i}, \quad \nabla _y u_{i,j+1}=\frac{u_{i,j+1}-u_{i,j}}{y_{j+1}-y_j}. \end{aligned}$$
Furthermore, we denote the constant \(C_{inv}>0\) satisfying the inverse inequality
$$\begin{aligned} ||\nabla _x u|| \le C_{inv}h_x^{-1}||u||, \quad ||\nabla _y u|| \le C_{inv}h_y^{-1}||u||, \end{aligned}$$
(40)
for any energy norm defined earlier.
3.1 The stability analysis
Theorem 3.1
Assume that the time step size \(\tau \) satisfies the constraint
$$\begin{aligned} \tau \le \min \left( \frac{C_{inv}h_y}{2C_v}, \frac{C_{inv}h_x}{2C_v}, \frac{1}{2\omega _{pe}}, \frac{1}{2\omega _{pm}}\right) , \end{aligned}$$
(41)
then the solution of the fully discrete scheme (34)–(39) satisfies the following stability: For any \(1\le n\le N_t\),
$$\begin{aligned}&\epsilon _0\left( ||E_x^{n+1}||_E^2+||E_y^{n+1}||_E^2\right) +\mu _0||H^{n+\frac{3}{2}}||^2_H\nonumber \\&\qquad +\frac{1}{\epsilon _0\omega _{pe}^2} \left( \left| \left| J_x^{n+\frac{3}{2}}\right| \right| ^2_J +\left| \left| J_y^{n+\frac{3}{2}}\right| \right| ^2_J\right) +\frac{1}{\mu _0\omega _{pm}^2}||K^{n+2}||^2_K \nonumber \\&\quad \le C\bigg [\epsilon _0 \left( \left| \left| E_x^{0}\right| \right| _E^2 +\left| \left| E_y^{0}\right| \right| _E^2\right) +\mu _0||H^{\frac{1}{2}}||^2_H\nonumber \\&\qquad +\frac{1}{\epsilon _0\omega _{pe}^2}\left( ||J_x^{\frac{1}{2}}||^2_J +\left| \left| J_y^{\frac{1}{2}}\right| \right| ^2_J\right) +\frac{1}{\mu _0\omega _{pm}^2}||K^{1}||^2_K\bigg ], \end{aligned}$$
(42)
where the constant \(C>0\) is independent of \(\tau , h_x\) and \(h_y.\)
Proof
Multiplying (34) by \(\tau |T_{i,j-\frac{1}{2}}|(E_{x,i+\frac{1}{2},j}^{n+1}+E_{x,i+\frac{1}{2},j}^{n})\), (35) by \(\tau |T_{i-\frac{1}{2},j}|(E_{y,i,j+\frac{1}{2}}^{n+1}+E_{y,i,j+\frac{1}{2}}^{n})\), (36) by \(\tau |T_{ij}|(H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{3}{2}} +H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}})\), (37) by \(\tau |T_{i,j-\frac{1}{2}}|(J_{x,i+\frac{1}{2},j}^{n+\frac{3}{2}} +J_{x,i+\frac{1}{2},j}^{n+\frac{1}{2}})\), (38) by \(\tau |T_{i-\frac{1}{2},j}|(J_{y,i,j+\frac{1}{2}}^{n+\frac{3}{2}} +J_{y,i,j+\frac{1}{2}}^{n+\frac{1}{2}})\), (39) by \(\tau |T_{ij}|(K_{i+\frac{1}{2},j+\frac{1}{2}}^{n+2} +K_{i+\frac{1}{2},j+\frac{1}{2}}^{n+1})\), then summing up the results, we obtain the sum of the right hand side as
$$\begin{aligned} RHS= & {} \tau \sum _{\begin{array}{c} 0\le i \le N_x-1 \\ 1\le j \le N_y-1 \end{array}}\left[ (x_{i+1}-x_i)\left( H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}}-H_{i+\frac{1}{2},j-\frac{1}{2}}^{n+\frac{1}{2}}\right) \right. \\&\left. -\,J_{x,i+\frac{1}{2},j}^{n+\frac{1}{2}}\left| T_{i,j-\frac{1}{2}}\right| \right] \left( E_{x,i+\frac{1}{2},j}^{n+1}+E_{x,i+\frac{1}{2},j}^{n}\right) \nonumber \\&+\,\tau \sum _{\begin{array}{c} 1\le i \le N_x-1 \\ 0\le j \le N_y-1 \end{array}} \left[ -(y_{j+1}-y_j)\left( H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}}-H_{i-\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}}\right) \right. \\&\left. -\,J_{y,i,j+\frac{1}{2}}^{n+\frac{1}{2}}\left| T_{i-\frac{1}{2},j}\right| \right] \left( E_{y,i,j+\frac{1}{2}}^{n+1}+E_{y,i,j+\frac{1}{2}}^{n}\right) \nonumber \\&+\, \tau \sum _{\begin{array}{c} 0\le i \le N_x-1 \\ 0\le j \le N_y-1 \end{array}} \left[ -(y_{j+1}-y_j)\left( E_{y,i+1,j+\frac{1}{2}}^{n+1}-E_{y,i,j+\frac{1}{2}}^{n+1}\right) \right. \\&+\,(x_{i+1}-x_i)\left( E_{x,i+\frac{1}{2},j+1}^{n+1}-E_{x,i+\frac{1}{2},j}^{n+1}\right) \nonumber \\&\left. -\,K_{i+\frac{1}{2},j+\frac{1}{2}}^{n+1}|T_{ij}|\right] \left( H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{3}{2}}+H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}}\right) \nonumber \\&+\,\tau \sum _{\begin{array}{c} 0\le i \le N_x-1 \\ 1\le j \le N_y-1 \end{array}}E_{x,i+\frac{1}{2},j}^{n+1} \cdot \left| T_{i,j-\frac{1}{2}}\right| \left( J_{x,i+\frac{1}{2},j}^{n+\frac{3}{2}}+J_{x,i+\frac{1}{2},j}^{n+\frac{1}{2}}\right) \nonumber \\&+\,\tau \sum _{\begin{array}{c} 1\le i \le N_x-1 \\ 0\le j \le N_y-1 \end{array}}E_{y,i,j+\frac{1}{2}}^{n+1} \cdot \left| T_{i-\frac{1}{2},j}\right| \left( J_{y,i,j+\frac{1}{2}}^{n+\frac{3}{2}}+J_{y,i,j+\frac{1}{2}}^{n+\frac{1}{2}}\right) \nonumber \\&+\,\tau \sum _{\begin{array}{c} 0\le i \le N_x-1 \\ 0\le j \le N_y-1 \end{array}}H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{3}{2}} \cdot |T_{ij}|\left( K_{i+\frac{1}{2},j+\frac{1}{2}}^{n+2}+K_{i+\frac{1}{2},j+\frac{1}{2}}^{n+1}\right) . \end{aligned}$$
Regrouping those terms in RHS, we rewrite RHS as
$$\begin{aligned} RHS= & {} \tau \sum _{0\le i \le N_x-1}(x_{i+1}-x_i)\sum _{1\le j \le N_y-1} \left[ \left( H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}}-H_{i+\frac{1}{2},j-\frac{1}{2}}^{n+\frac{1}{2}}\right) \right. \nonumber \\&\times \left. \left( E_{x,i+\frac{1}{2},j}^{n+1}+E_{x,i+\frac{1}{2},j}^{n}\right) +\left( E_{x,i+\frac{1}{2},j+1}^{n+1} -E_{x,i+\frac{1}{2},j}^{n+1}\right) \right. \nonumber \\&\left. \times \left( H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{3}{2}} +H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}}\right) \right] \nonumber \\ \end{aligned}$$
$$\begin{aligned}&+\,\tau \sum _{0\le j \le N_y-1}(y_{j+1}-y_j)\sum _{1\le i \le N_x-1} \left[ \left( H_{i-\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}} -H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}}\right) \right. \nonumber \\&\times \left. \left( E_{y,i,j+\frac{1}{2}}^{n+1}+E_{y,i,j+\frac{1}{2}}^{n}\right) +\left( E_{y,i,j+\frac{1}{2}}^{n+1}-E_{y,i+1,j+\frac{1}{2}}^{n+1}\right) \right. \nonumber \\&\times \left. \left( H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{3}{2}}+H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}}\right) \right] \nonumber \\&+\,\tau \sum _{\begin{array}{c} 0\le i \le N_x-1 \\ 1\le j \le N_y-1 \end{array}} \left| T_{i,j-\frac{1}{2}}\right| \left[ -J_{x,i+\frac{1}{2},j}^{n+\frac{1}{2}} \left( E_{x,i+\frac{1}{2},j}^{n+1}+E_{x,i+\frac{1}{2},j}^{n}\right) \right. \nonumber \\&\left. +\,\left( J_{x,i+\frac{1}{2},j}^{n+\frac{3}{2}} +J_{x,i+\frac{1}{2},j}^{n+\frac{1}{2}}\right) E_{x,i+\frac{1}{2},j}^{n+1}\right] \nonumber \\&+\,\tau \sum _{\begin{array}{c} 1\le i \le N_x-1 \\ 0\le j \le N_y-1 \end{array}}\left| T_{i-\frac{1}{2},j}\right| \left[ -J_{y,i,j+\frac{1}{2}}^{n+\frac{1}{2}} \left( E_{y,i,j+\frac{1}{2}}^{n+1}+E_{y,i,j+\frac{1}{2}}^{n}\right) \right. \nonumber \\ \end{aligned}$$
$$\begin{aligned}&\left. +\,\left( J_{y,i,j+\frac{1}{2}}^{n+\frac{3}{2}} +J_{y,i,j+\frac{1}{2}}^{n+\frac{1}{2}}\right) E_{y,i,j+\frac{1}{2}}^{n+1}\right] \nonumber \\&+\,\tau \sum _{\begin{array}{c} 0\le i \le N_x-1 \\ 0\le j \le N_y-1 \end{array}}|T_{ij}| \left[ -K_{i+\frac{1}{2},j+\frac{1}{2}}^{n+1} \left( H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{3}{2}} +H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}}\right) \right. \nonumber \\ \end{aligned}$$
$$\begin{aligned}&\left. +\,H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{3}{2}} \left( K_{i+\frac{1}{2},j+\frac{1}{2}}^{n+2} +K_{i+\frac{1}{2},j+\frac{1}{2}}^{n+1}\right) \right] \nonumber \\:= & {} \tau \left[ \sum _{0\le i \le N_x-1}(x_{i+1}-x_i)R_1 + \sum _{0\le j \le N_y-1}(y_{j+1}-y_j)R_2 + R_3 +R_4 +R_5 \right] . \nonumber \\ \end{aligned}$$
(43)
To evaluate the above RHS, below we evaluate each term separately. First, note that
$$\begin{aligned} \sum _{n=0}^{N_t}R_1= & {} \sum _{n=0}^{N_t}\sum _{0\le j\le N_y-1} \left[ \left( H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}}-H_{i+\frac{1}{2},j-\frac{1}{2}}^{n+\frac{1}{2}})(E_{x,i+\frac{1}{2},j}^{n+1}+E_{x,i+\frac{1}{2},j}^{n}\right) \right. \nonumber \\&\left. +\left( E_{x,i+\frac{1}{2},j+1}^{n+1}-E_{x,i+\frac{1}{2},j}^{n+1}\right) \left( H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{3}{2}}+H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}}\right) \right] \nonumber \\= & {} \sum _{n=0}^{N_t}\sum _{0\le j\le N_y-1}\left[ \left( H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}}E_{x,i+\frac{1}{2},j}^{n}-H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{3}{2}}E_{x,i+\frac{1}{2},j}^{n+1}\right) \right. \nonumber \\&\left. +\left( H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}}E_{x,i+\frac{1}{2},j+1}^{n+1} -H_{i+\frac{1}{2},j-\frac{1}{2}}^{n+\frac{1}{2}}E_{x,i+\frac{1}{2},j}^{n+1}\right) \right] \nonumber \\&+\sum _{n=0}^{N_t}\sum _{0\le j\le N_y-1} \left[ \left( H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{3}{2}}E_{x,i+\frac{1}{2},j+1}^{n+1}-H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}}E_{x,i+\frac{1}{2},j+1}^{n}\right) \right. \nonumber \\&\left. + \left( H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}}E_{x,i+\frac{1}{2},j+1}^{n}-H_{i+\frac{1}{2},j-\frac{1}{2}}^{n+\frac{1}{2}}E_{x,i+\frac{1}{2},j}^{n}\right) \right] \nonumber \\ \end{aligned}$$
$$\begin{aligned}= & {} \sum _{0\le j\le N_y-1} \left( H_{i+\frac{1}{2},j+\frac{1}{2}}^{\frac{1}{2}}E_{x,i+\frac{1}{2},j}^{0} -H_{i+\frac{1}{2},j+\frac{1}{2}}^{N_t+\frac{3}{2}}E_{x,i+\frac{1}{2},j}^{N_t+1}\right) \nonumber \\&+\sum _{n=0}^{N_t}\left( H_{i+\frac{1}{2},N_y+\frac{1}{2}}^{n+\frac{1}{2}}E_{x,i+\frac{1}{2},N_y}^{n+1} -H_{i+\frac{1}{2},-\frac{1}{2}}^{n+\frac{1}{2}}E_{x,i+\frac{1}{2},0}^{n+1}\right) \nonumber \\&+ \sum _{0\le j\le N_y-1} \left( H_{i+\frac{1}{2},j+\frac{1}{2}}^{N_t+\frac{3}{2}}E_{x,i+\frac{1}{2},j+1}^{N_t+1}-H_{i+\frac{1}{2},j+\frac{1}{2}}^{\frac{1}{2}}E_{x,i+\frac{1}{2},j+1}^{0}\right) \nonumber \\&+ \sum _{n=0}^{N_t}(H_{i+\frac{1}{2},N_y+\frac{1}{2}}^{n+\frac{1}{2}}E_{x,i+\frac{1}{2},N_y}^{n}) -H_{i+\frac{1}{2},-\frac{1}{2}}^{n+\frac{1}{2}}E_{x,i+\frac{1}{2},0}^{n}) \nonumber \\ \end{aligned}$$
$$\begin{aligned}= & {} \sum _{0\le j\le N_y-1} \left( H_{i+\frac{1}{2},j+\frac{1}{2}}^{\frac{1}{2}}E_{x,i+\frac{1}{2},j}^{0} -H_{i+\frac{1}{2},j+\frac{1}{2}}^{N_t+\frac{3}{2}}E_{x,i+\frac{1}{2},j}^{N_t+1}\right) \nonumber \\&+ \sum _{0\le j\le N_y-1} \left( H_{i+\frac{1}{2},j+\frac{1}{2}}^{N_t+\frac{3}{2}}E_{x,i+\frac{1}{2},j+1}^{N_t+1}-H_{i+\frac{1}{2},j+\frac{1}{2}}^{\frac{1}{2}}E_{x,i+\frac{1}{2},j+1}^{0}\right) \nonumber \\= & {} \sum _{0\le j\le N_y-1}(y_{j+1}-y_j) \left( H_{i+\frac{1}{2},j+\frac{1}{2}}^{N_t+\frac{3}{2}}\nabla _yE_{x,i+\frac{1}{2},j+1}^{N_t+1} -H_{i+\frac{1}{2},j+\frac{1}{2}}^{\frac{1}{2}}\nabla _yE_{x,i+\frac{1}{2},j+1}^{0}\right) , \nonumber \\ \end{aligned}$$
(44)
where we used the PEC boundary condition (19) in the second last step, and the backward difference operator \(\nabla _y\) in the last step. Note that in the first step, we extended the original sum of \(1\le j\le N_y-1\) to \(0\le j\le N_y-1\). Even though \(H_{i+\frac{1}{2},-\frac{1}{2}}^{n+\frac{1}{2}}\) has subindex out of the original bound, its product with \(E_{x,i+\frac{1}{2},0}^{n+1}+E_{x,i+\frac{1}{2},0}^{n}=0\) (by the PEC boundary condition (19)) is still zero.
The term \(R_2\) can be evaluated as follows:
$$\begin{aligned} \sum _{n=0}^{N_t}R_2= & {} \sum _{n=0}^{N_t}\sum _{0\le i\le N_x-1} \left[ \left( H_{i-\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}}-H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}}\right) \left( E_{y,i,j+\frac{1}{2}}^{n+1}+E_{y,i,j+\frac{1}{2}}^{n}\right) \right. \nonumber \\&\left. +\left( E_{y,i,j+\frac{1}{2}}^{n+1}-E_{y,i+1,j+\frac{1}{2}}^{n+1}\right) \left( H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{3}{2}}+H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}}\right) \right] \nonumber \\= & {} \sum _{n=0}^{N_t}\sum _{0\le i\le N_x-1} \left[ \left( H_{i-\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}}E_{y,i,j+\frac{1}{2}}^{n+1}-H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}}E_{y,i+1,j+\frac{1}{2}}^{n+1}\right) \right. \nonumber \\&\left. + \left( -H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}}E_{y,i,j+\frac{1}{2}}^{n} +H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{3}{2}}E_{y,i,j+\frac{1}{2}}^{n+1}\right) \right] \nonumber \\&+ \sum _{n=0}^{N_t}\sum _{0\le i\le N_x-1}\left[ \left( H_{i-\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}}E_{y,i,j+\frac{1}{2}}^{n}-H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}}E_{y,i+1,j+\frac{1}{2}}^{n}\right) \right. \nonumber \\ \end{aligned}$$
$$\begin{aligned}&\left. + \left( H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}}E_{y,i+1,j+\frac{1}{2}}^{n} -H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{3}{2}}E_{y,i+1,j+\frac{1}{2}}^{n+1}\right) \right] \nonumber \\= & {} \sum _{n=0}^{N_t}\left( H_{-\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}}E_{y,0,j+\frac{1}{2}}^{n+1} -H_{N_x-\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}}E_{y,N_x,j+\frac{1}{2}}^{n+1}\right) \nonumber \\&+ \sum _{0\le i\le N_x-1}\left( -H_{i+\frac{1}{2},j+\frac{1}{2}}^{\frac{1}{2}}E_{y,i,j+\frac{1}{2}}^{0} +H_{i+\frac{1}{2},j+\frac{1}{2}}^{N_t+\frac{3}{2}}E_{y,i,j+\frac{1}{2}}^{N_t+1}\right) \nonumber \\&+\sum _{n=0}^{N_t}\left( H_{-\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}}E_{y,0,j+\frac{1}{2}}^{n} -H_{N_x+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}}E_{y,N_x,j+\frac{1}{2}}^{n}\right) \nonumber \\ \end{aligned}$$
$$\begin{aligned}&+ \sum _{0\le i\le N_x-1}\left( H_{i+\frac{1}{2},j+\frac{1}{2}}^{\frac{1}{2}}E_{y,i+1,j+\frac{1}{2}}^{0} -H_{i+\frac{1}{2},j+\frac{1}{2}}^{N_t+\frac{3}{2}}E_{y,i+1,j+\frac{1}{2}}^{N_t+1}\right) \nonumber \\= & {} \sum _{0\le i\le N_x-1}\left( -H_{i+\frac{1}{2},j+\frac{1}{2}}^{\frac{1}{2}}E_{y,i,j+\frac{1}{2}}^{0} +H_{i+\frac{1}{2},j+\frac{1}{2}}^{N_t+\frac{3}{2}}E_{y,i,j+\frac{1}{2}}^{N_t+1}\right) \nonumber \\&+ \sum _{0\le i\le N_x-1}\left( H_{i+\frac{1}{2},j+\frac{1}{2}}^{\frac{1}{2}}E_{y,i+1,j+\frac{1}{2}}^{0} -H_{i+\frac{1}{2},j+\frac{1}{2}}^{N_t+\frac{3}{2}}E_{y,i+1,j+\frac{1}{2}}^{N_t+1}\right) \nonumber \\= & {} \sum _{0\le i\le N_x-1}(x_{i+1}-x_i)\left( -H_{i+\frac{1}{2},j+\frac{1}{2}}^{N_t+\frac{3}{2}} \nabla _xE_{y,i+1,j+\frac{1}{2}}^{N_t+1} +H_{i+\frac{1}{2},j+\frac{1}{2}}^{\frac{1}{2}}\nabla _xE_{y,i+1,j+\frac{1}{2}}^{0}\right) , \nonumber \\ \end{aligned}$$
(45)
where the PEC boundary condition (19) was used in the second last step, and the backward difference operator \(\nabla _x\) was used in the last step. Here similarly to \(R_1\), in the first step we extended the original sum of \(1\le i\le N_x-1\) to \(0\le i\le N_x-1\). Even though \(H_{-\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}}\) has subindex out of the original bound, its product with \(E_{y,0,j+\frac{1}{2}}^{n+1}+E_{y,0,j+\frac{1}{2}}^{n}=0\) (by the PEC boundary condition (19)) is still zero.
Similarly, we can evaluate the rest terms in RHS (43) as follows.
$$\begin{aligned} \sum _{n=0}^{N_t}R_3= & {} \sum _{n=0}^{N_t}\sum _{\begin{array}{c} 0\le i \le N_x-1 \\ 0\le j \le N_y-1 \end{array}}\left| T_{i,j-\frac{1}{2}}\right| \left[ -J_{x,i+\frac{1}{2},j}^{n+\frac{1}{2}}\left( E_{x,i+\frac{1}{2},j}^{n+1}+E_{x,i+\frac{1}{2},j}^{n}\right) \right. \nonumber \\&\left. +\left( J_{x,i+\frac{1}{2},j}^{n+\frac{3}{2}}+J_{x,i+\frac{1}{2},j}^{n+\frac{1}{2}}\right) E_{x,i+\frac{1}{2},j}^{n+1}\right] \nonumber \\= & {} \sum _{\begin{array}{c} 0\le i \le N_x-1 \\ 0\le j \le N_y-1 \end{array}}\left| T_{i,j-\frac{1}{2}}\right| \left( J_{x,i+\frac{1}{2},j}^{N_t+\frac{3}{2}}E_{x,i+\frac{1}{2},j}^{N_t+1} -J_{x,i+\frac{1}{2},j}^{\frac{1}{2}}E_{x,i+\frac{1}{2},j}^{0}\right) , \end{aligned}$$
(46)
$$\begin{aligned} \sum _{n=0}^{N_t}R_4= & {} \sum _{n=0}^{N_t}\sum _{\begin{array}{c} 0\le i \le N_x-1 \\ 0\le j \le N_y-1 \end{array}}\left| T_{i-\frac{1}{2},j}\right| \left[ -J_{y,i,j+\frac{1}{2}}^{n+\frac{1}{2}}\left( E_{y,i,j+\frac{1}{2}}^{n+1}+E_{y,i,j+\frac{1}{2}}^{n}\right) \right. \nonumber \\&\left. +\left( J_{y,i,j+\frac{1}{2}}^{n+\frac{3}{2}}+J_{y,i,j+\frac{1}{2}}^{n+\frac{1}{2}}\right) E_{y,i,j+\frac{1}{2}}^{n+1}\right] \nonumber \\= & {} \sum _{\begin{array}{c} 0\le i \le N_x-1 \\ 0\le j \le N_y-1 \end{array}}\left| T_{i-\frac{1}{2},j}\right| \left( J_{y,i,j+\frac{1}{2}}^{N_t+\frac{3}{2}}E_{y,i,j+\frac{1}{2}}^{N_t+1} -J_{y,i,j+\frac{1}{2}}^{\frac{1}{2}}E_{y,i,j+\frac{1}{2}}^{0}\right) , \end{aligned}$$
(47)
and
$$\begin{aligned} \sum _{n=0}^{N_t}R_5= & {} \sum _{n=0}^{N_t}\sum _{\begin{array}{c} 0\le i \le N_x-1 \\ 0\le j \le N_y-1 \end{array}}|T_{ij}| \left[ -K_{i+\frac{1}{2},j+\frac{1}{2}}^{n+1}\left( H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{3}{2}} +H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}}\right) \right. \nonumber \\&\left. +H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{3}{2}}\left( K_{i+\frac{1}{2},j+\frac{1}{2}}^{n+2} +K_{i+\frac{1}{2},j+\frac{1}{2}}^{n+1}\right) \right] \nonumber \\= & {} \sum _{\begin{array}{c} 0\le i \le N_x-1 \\ 0\le j \le N_y-1 \end{array}}|T_{ij}| \left( H_{i+\frac{1}{2},j+\frac{1}{2}}^{N_t+\frac{3}{2}}K_{i+\frac{1}{2},j+\frac{1}{2}}^{N_t+2} -H_{i+\frac{1}{2},j+\frac{1}{2}}^{\frac{1}{2}}K_{i+\frac{1}{2},j+\frac{1}{2}}^{1}\right) . \end{aligned}$$
(48)
Summing up (43) from \(n=0\) to \(N_t\), then substituting the estimates (44)–(48), and using the energy norm notations, we have
$$\begin{aligned}&\epsilon _0\left( \left| \left| E_x^{N_t+1}\right| \right| ^2_E-\left| \left| E_x^{0} \right| \right| ^2_E\right) +\epsilon _0\left( \left| \left| E_y^{N_t+1}\right| \right| ^2_E -\left| \left| E_y^{0}\right| \right| ^2_E\right) \nonumber \\&\qquad +\,\mu _0\left( ||H^{N_t+\frac{3}{2}}||^2_H-||H^{\frac{1}{2}}||^2_H\right) \nonumber \\&\qquad +\, \frac{1}{\epsilon _0\omega ^2_{pe}} \left( \left| \left| J_x^{N_t+\frac{3}{2}}\right| \right| ^2_J -\left| \left| J_x^{\frac{1}{2}}\right| \right| ^2_J\right) + \frac{1}{\epsilon _0\omega ^2_{pe}} \left( \left| \left| J_y^{N_t+\frac{3}{2}}\right| \right| ^2_J -\left| \left| J_y^{\frac{1}{2}}\right| \right| ^2_J\right) \nonumber \\&\qquad + \,\frac{1}{\mu _0\omega ^2_{pm}} \left( ||K^{N_t+2}||^2_K-||K^{1}||^2_K\right) \nonumber \\&\quad \le \tau \sum _{\begin{array}{c} 0\le i \le N_x-1 \\ 0\le j \le N_y-1 \end{array}}|T_{ij}|\left( H_{i+\frac{1}{2},j+\frac{1}{2}}^{N_t+\frac{3}{2}}\nabla _yE_{x,i+\frac{1}{2},j+1}^{N_t+1} -H_{i+\frac{1}{2},j+\frac{1}{2}}^{\frac{1}{2}}\nabla _yE_{x,i+\frac{1}{2},j+1}^{0}\right) \nonumber \\&\qquad +\,\tau \sum _{\begin{array}{c} 0\le i \le N_x-1 \\ 0\le j \le N_y-1 \end{array}}|T_{ij}| \left( -H_{i+\frac{1}{2},j+\frac{1}{2}}^{N_t+\frac{3}{2}} \nabla _xE_{y,i+1,j+\frac{1}{2}}^{N_t+1} +H_{i+\frac{1}{2},j+\frac{1}{2}}^{\frac{1}{2}}\nabla _xE_{y,i+1,j+\frac{1}{2}}^{0}\right) \nonumber \\&\qquad +\, \tau \sum _{\begin{array}{c} 0\le i \le N_x-1 \\ 0\le j \le N_y-1 \end{array}}\left| T_{i,j-\frac{1}{2}}\right| \left( J_{x,i+\frac{1}{2},j}^{N_t+\frac{3}{2}}E_{x,i+\frac{1}{2},j}^{N_t+1} -J_{x,i+\frac{1}{2},j}^{\frac{1}{2}}E_{x,i+\frac{1}{2},j}^{0}\right) \nonumber \\&\qquad +\,\tau \sum _{\begin{array}{c} 0\le i \le N_x-1 \\ 0\le j \le N_y-1 \end{array}}\left| T_{i-\frac{1}{2},j}\right| \left( J_{y,i,j+\frac{1}{2}}^{N_t+\frac{3}{2}}E_{y,i,j+\frac{1}{2}}^{N_t+1} -J_{y,i,j+\frac{1}{2}}^{\frac{1}{2}}E_{y,i,j+\frac{1}{2}}^{0}\right) \nonumber \\&\qquad +\,\tau \sum _{\begin{array}{c} 0\le i \le N_x-1 \\ 0\le j \le N_y-1 \end{array}}|T_{ij}| \left( H_{i+\frac{1}{2},j+\frac{1}{2}}^{N_t+\frac{3}{2}}K_{i+\frac{1}{2},j+\frac{1}{2}}^{N_t+2} -H_{i+\frac{1}{2},j+\frac{1}{2}}^{\frac{1}{2}}K_{i+\frac{1}{2},j+\frac{1}{2}}^{1}\right) . \end{aligned}$$
(49)
Now we just need to bound those right hand side terms of (49). Using the Cauchy–Schwarz inequality and the inverse estimate (40), we have
$$\begin{aligned}&\tau \sum _{\begin{array}{c} 0\le i \le N_x-1 \\ 0\le j \le N_y-1 \end{array}}|T_{ij}|\cdot H_{i+\frac{1}{2},j+\frac{1}{2}}^{N_t+\frac{3}{2}}\nabla _yE_{x,i+\frac{1}{2},j+1}^{N_t+1} \nonumber \\&\quad \le \tau \left( \sum _{\begin{array}{c} 0\le i \le N_x-1 \\ 0\le j \le N_y-1 \end{array}}|T_{ij}|\cdot |H_{i+\frac{1}{2},j+\frac{1}{2}}^{N_t+\frac{3}{2}}|^2\right) ^{1/2} \left( \sum _{\begin{array}{c} 0\le i \le N_x-1 \\ 0\le j \le N_y-1 \end{array}}|T_{ij}| \cdot |\nabla _yE_{x,i+\frac{1}{2},j+1}^{N_t+1}|^2\right) ^{1/2} \nonumber \\&\quad = \tau ||H^{N_t+\frac{3}{2}}||_H||\nabla _yE_x^{N_t+1}||_E \le \delta \mu _0||H^{N_t+\frac{3}{2}}||_H^2\nonumber \\&\qquad +\frac{1}{4\delta }\cdot \frac{\left( \tau C_{inv}h_y^{-1}\right) ^2}{\mu _0\epsilon _0}\cdot \epsilon _0\left| \left| E_x^{N_t+1}\right| \right| _E^2. \end{aligned}$$
(50)
Similarly, we can obtain
$$\begin{aligned}&\tau \sum _{\begin{array}{c} 0\le i \le N_x-1 \\ 0\le j \le N_y-1 \end{array}}|T_{ij}|\cdot H_{i+\frac{1}{2},j+\frac{1}{2}}^{N_t+\frac{3}{2}}\nabla _xE_{y,i+1,j+\frac{1}{2}}^{N_t+1} \le \delta \mu _0||H^{N_t+\frac{3}{2}}||_H^2\nonumber \\&\quad +\frac{1}{4\delta }\cdot \frac{(\tau C_{inv}h_x^{-1})^2}{\mu _0\epsilon _0}\cdot \epsilon _0||E_y^{N_t+1}||_E^2. \end{aligned}$$
(51)
By the similar technique, we can prove that
$$\begin{aligned}&\tau \sum _{\begin{array}{c} 0\le i \le N_x-1 \\ 1\le j \le N_y-1 \end{array}}\left| T_{i,j-\frac{1}{2}}\right| \cdot J_{x,i+\frac{1}{2},j}^{N_t+\frac{3}{2}}E_{x,i+\frac{1}{2},j}^{N_t+1} \nonumber \\&\quad \le \tau ||J_x^{N_t+\frac{3}{2}}||_J||E_x^{N_t+1}||_E \le \frac{\tau \omega _{pe}}{2}\left( \frac{1}{\epsilon _0\omega ^2_{pe}}||J_x^{N_t+\frac{3}{2}}||_J^2 +\epsilon _0||E_x^{N_t+1}||_E^2\right) , \qquad \end{aligned}$$
(52)
$$\begin{aligned}&\tau \sum _{\begin{array}{c} 1\le i \le N_x-1 \\ 0\le j \le N_y-1 \end{array}}\left| T_{i-\frac{1}{2},j}\right| \cdot J_{y,i,j+\frac{1}{2}}^{N_t+\frac{3}{2}}E_{y,i,j+\frac{1}{2}}^{N_t+1} \nonumber \\&\quad \le \tau ||J_y^{N_t+\frac{3}{2}}||_J||E_y^{N_t+1}||_E \le \frac{\tau \omega _{pe}}{2}\left( \frac{1}{\epsilon _0\omega ^2_{pe}}||J_y^{N_t+\frac{3}{2}}||_J^2 +\epsilon _0||E_y^{N_t+1}||_E^2\right) , \end{aligned}$$
(53)
and
$$\begin{aligned}&\tau \sum _{\begin{array}{c} 0\le i \le N_x-1 \\ 0\le j \le N_y-1 \end{array}}|T_{ij}|\cdot H_{i+\frac{1}{2},j+\frac{1}{2}}^{N_t+\frac{3}{2}}K_{i+\frac{1}{2},j+\frac{1}{2}}^{N_t+2} \nonumber \\&\quad \le \tau ||H^{N_t+\frac{3}{2}}||_H||K^{N_t+2}||_K \le \frac{\tau \omega _{pm}}{2}\left( \mu _0||H^{N_t+\frac{3}{2}}||_H^2 +\frac{1}{\mu _0\omega ^2_{pm}}||K^{N_t+2}||_K^2\right) . \nonumber \\ \end{aligned}$$
(54)
Substituting the estimates (50)–(54) into (49), then choosing \(\delta \) and \(\tau \) small enough so that the left hand side terms of (49) can control those corresponding terms on the right hand side. A specific choice can be
$$\begin{aligned} \delta =\frac{1}{4}, ~\tau \le \frac{C_{inv}h_y}{2C_v}, ~\tau \le \frac{C_{inv}h_x}{2C_v}, ~\tau \le \frac{1}{2\omega _{pe}},~\tau \le \frac{1}{2\omega _{pm}}. \end{aligned}$$
This completes the proof. \(\square \)
3.2 The error estimate
To make the error analysis easy to follow, we denote the errors by their corresponding script letters. For example, the error of \(E_x\) at point \((x_{i+\frac{1}{2}},y_j,t_n)\) is denoted by \(\mathcal {E}^n_{x,i+\frac{1}{2},j}=E_x(x_{i+\frac{1}{2}},y_j,t_n)-E^n_{x,i+\frac{1}{2},j}\), where \(E_x(x_{i+\frac{1}{2}},y_j,t_n)\) and \(E^n_{x,i+\frac{1}{2},j}\) denote the exact and numerical solutions of \(E_x\) at point \((x_{i+\frac{1}{2}},y_j,t_n)\), respectively. Similar error notations given below will be used for other variables:
$$\begin{aligned} \mathcal {E}^n_{y,i,j+\frac{1}{2}},~~\mathcal {H}^{n+\frac{1}{2}}_{i+\frac{1}{2},j+\frac{1}{2}}, ~~\mathcal {J}^{n+\frac{1}{2}}_{x,i+\frac{1}{2},j},~~\mathcal {J}^{n+\frac{1}{2}}_{y,i,j+\frac{1}{2}},~~\mathcal {K}^{n+1}_{i+\frac{1}{2},j+\frac{1}{2}}. \end{aligned}$$
3.2.1 The error equation for \(E_x\)
Multiplying (34) by \(|T_{i,j-\frac{1}{2}}|\) (the area of rectangle \(T_{i,j-\frac{1}{2}}\)), we can rewrite (34) as follows:
$$\begin{aligned} \frac{\epsilon _0\left| T_{i,j-\frac{1}{2}}\right| }{\tau } \left( E_{x,i+\frac{1}{2},j}^{n+1}-E_{x,i+\frac{1}{2},j}^{n}\right)= & {} (x_{i+1}-x_i)\left( H_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}} -H_{i+\frac{1}{2},j-\frac{1}{2}}^{n+\frac{1}{2}}\right) \\&-\left| T_{i,j-\frac{1}{2}}\right| J_{x,i+\frac{1}{2},j}^{n+\frac{1}{2}}, \end{aligned}$$
from which we can easily obtain the error equation for \(E_x\):
$$\begin{aligned} \frac{\epsilon _0\left| T_{i,j-\frac{1}{2}}\right| }{\tau } \left( \mathcal {E}_{x,i+\frac{1}{2},j}^{n+1}-\mathcal {E}_{x,i+\frac{1}{2},j}^{n}\right)= & {} (x_{i+1}-x_i)\left( \mathcal {H}_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}} -\mathcal {H}_{i+\frac{1}{2},j-\frac{1}{2}}^{n+\frac{1}{2}}\right) \nonumber \\&-\left| T_{i,j-\frac{1}{2}}\right| \mathcal {J}_{x,i+\frac{1}{2},j}^{n+\frac{1}{2}}+R_1, \end{aligned}$$
(55)
where the local truncation error term \(R_1\) is given by
$$\begin{aligned} R_1= & {} \frac{\epsilon _0\left| T_{i,j-\frac{1}{2}}\right| }{\tau } \left( E_x\left( x_{i+\frac{1}{2}},y_j,t_{n+1}\right) -E_x\left( x_{i+\frac{1}{2}},y_j,t_n\right) \right) \nonumber \\&-\,(x_{i+1}-x_i) \left( H\left( x_{i+\frac{1}{2}},y_{j+\frac{1}{2}},t_{n+\frac{1}{2}}\right) -H\left( x_{i+\frac{1}{2}},y_{j-\frac{1}{2}},t_{n+\frac{1}{2}}\right) \right) \nonumber \\&+\left| T_{i,j-\frac{1}{2}}\right| J_x\left( x_{i+\frac{1}{2}},y_j,t_{n+\frac{1}{2}}\right) . \end{aligned}$$
(56)
Integrating (7) from \(t=t_n\) to \(t_{n+1}\) and dividing the resultant by \(\tau \), we have
$$\begin{aligned}&\frac{\epsilon _0}{\tau }\iint _{T_{i,j-\frac{1}{2}}}(E_x(x,y,t_{n+1})-E_x(x,y,t_{n}))dxdy \nonumber \\&\quad =\frac{1}{\tau }\int _{t_n}^{t_{n+1}}\int _{x_i}^{x_{i+1}} \left( H\left( x,y_{j+\frac{1}{2}},t\right) -H\left( x,y_{j-\frac{1}{2}},t\right) \right) dxdt\nonumber \\&\qquad -\frac{1}{\tau }\int _{t_n}^{t_{n+1}} \iint _{T_{i,j-\frac{1}{2}}}J_x(x,y,t)dxdydt. \end{aligned}$$
(57)
Subtracting (57) from (56), we can rewrite \(R_1\) as follows:
$$\begin{aligned} R_1= & {} \frac{\epsilon _0}{\tau }\iint _{T_{i,j-\frac{1}{2}}} \left[ \left( E_x\left( x_{i+\frac{1}{2}},y_j,t_{n+1}\right) -E_x(x,y,t_{n+1})\right) \right. \nonumber \\&\left. -\left( E_x\left( x_{i+\frac{1}{2}},y_j,t_n\right) -E_x(x,y,t_{n})\right) \right] dxdy \nonumber \\&-\left\{ \int _{x_i}^{x_{i+1}} \left( H\left( x_{i+\frac{1}{2}},y_{j+\frac{1}{2}},t_{n+\frac{1}{2}}\right) -H\left( x_{i+\frac{1}{2}},y_{j-\frac{1}{2}},t_{n+\frac{1}{2}}\right) \right) dx \right. \nonumber \\&\left. -\frac{1}{\tau }\int _{t_n}^{t_{n+1}}\int _{x_i}^{x_{i+1}} \left( H\left( x,y_{j+\frac{1}{2}},t\right) -H\left( x,y_{j-\frac{1}{2}},t\right) \right) dxdt \right\} \nonumber \\&+\left[ \iint _{T_{i,j-\frac{1}{2}}} J_x\left( x_{i+\frac{1}{2}},y_j,t_{n+\frac{1}{2}}\right) dxdy -\frac{1}{\tau }\int _{t_n}^{t_{n+1}}\iint _{T_{i,j-\frac{1}{2}}}J_x(x,y,t)dxdydt\right] \nonumber \\= & {} R_{11} + R_{12} + R_{13}. \end{aligned}$$
(58)
Following the same technique used for deriving (29), for any function f we can prove that
$$\begin{aligned}&\iint _{T_{i,j-\frac{1}{2}}}\left( f(x,y,t_{n+1}) -f\left( x_{i+\frac{1}{2}},y_j,t_{n+1}\right) \right) dxdy\nonumber \\&\qquad -\iint _{T_{i,j-\frac{1}{2}}}\left( f(x,y,t_{n}) -f\left( x_{i+\frac{1}{2}},y_j,t_{n}\right) \right) dxdy \nonumber \\&\quad = \iint _{T_{i,j-\frac{1}{2}}}\left[ \frac{1}{2}\left( x-x_{i+\frac{1}{2}}\right) ^2\left( \frac{\partial ^2 f}{\partial x^2}(q_1,t_{n+1})-\frac{\partial ^2 f}{\partial x^2}(q_1,t_{n})\right) \right. \nonumber \\&\left. \qquad + \frac{1}{2}(y-y_j)^2\left( \frac{\partial ^2 f}{\partial y^2}(q_2,t_{n+1})-\frac{\partial ^2 f}{\partial y^2}(q_2,t_{n})\right) \right] dxdy \nonumber \\&\quad = \tau \iint _{T_{i,j-\frac{1}{2}}}\left[ \frac{1}{2} \left( x-x_{i+\frac{1}{2}}\right) ^2\frac{\partial ^3 f}{\partial t\partial x^2}(q_1,t_*) + \frac{1}{2}(y-y_j)^2\frac{\partial ^3 f}{\partial t\partial y^2}(q_2,t_*)\right] dxdy, \nonumber \\ \end{aligned}$$
(59)
where we denote \(q_1\) and \(q_2\) for some points between \((x_{i+\frac{1}{2}},y_j)\) and (x, y), and \(t_*\) for some point between \(t_n\) and \(t_{n+1}\). In the last step we used the following Taylor expansion
$$\begin{aligned} g(t_{n+1})-g(t_n)=\tau \frac{\partial g}{\partial t}(t_*) \end{aligned}$$
with \(g=\frac{\partial ^2 f}{\partial x^2}\) and \(g=\frac{\partial ^2 f}{\partial y^2}\), respectively.
Applying (59) with \(f=E_x\), we can bound \(R_{11}\) as follows:
$$\begin{aligned} R_{11}= & {} \frac{\epsilon _0}{\tau }\iint _{T_{i,j -\frac{1}{2}}}\left[ \frac{1}{2} \left( x-x_{i+\frac{1}{2}}\right) ^2\tau \frac{\partial ^3 E_x}{\partial t\partial x^2}(q_1,t_*) + \frac{1}{2}(y-y_j)^2\tau \frac{\partial ^3 E_x}{\partial t\partial y^2}(q_2,t_*)\right] dxdy \\= & {} \left( O\left( h_x^2\right) \left| \frac{\partial ^3 E_x}{\partial t\partial x^2}\right| _{\infty } + O\left( h_y^2\right) \left| \frac{\partial ^3 E_x}{\partial t\partial y^2}\right| _{\infty }\right) \left| T_{i,j-\frac{1}{2}}\right| . \end{aligned}$$
Similarly, by the Taylor expansion, we can estimate \(R_{12}\) as follows:
$$\begin{aligned} R_{12}= & {} -\int _{x_i}^{x_{i+1}}\int _{y_{j-\frac{1}{2}}}^{y_{j+\frac{1}{2}}}\frac{\partial H}{\partial y}\left( x_{i+\frac{1}{2}},y,t_{n+\frac{1}{2}}\right) dydx\nonumber \\&+\frac{1}{\tau }\int _{t_n}^{t_{n+1}}\int _{x_i}^{x_{i+1}}\int _{y_{j-\frac{1}{2}}}^{y_{j+\frac{1}{2}}}\frac{\partial H}{\partial y}(x,y,t)dydxdt \\= & {} -\int _{x_i}^{x_{i+1}}\int _{y_{j-\frac{1}{2}}}^{y_{j+\frac{1}{2}}} \left[ \frac{\partial H}{\partial y}\left( x_{i+\frac{1}{2}},y,t_{n+\frac{1}{2}}\right) -\frac{\partial H}{\partial y}(x,y,t_{n+\frac{1}{2}})\right] dydx \\&+ \int _{x_i}^{x_{i+1}}\int _{y_{j-\frac{1}{2}}}^{y_{j+\frac{1}{2}}} \frac{1}{\tau }\int _{t_n}^{t_{n+1}} \left[ \frac{\partial H}{\partial y}(x,y,t)-\frac{\partial H}{\partial y} \left( x,y,t_{n+\frac{1}{2}}\right) \right] dtdydx \\= & {} \iint _{T_{i,j-\frac{1}{2}}} \frac{1}{2} \left( x-x_{i+\frac{1}{2}}\right) ^2\frac{\partial ^3 H}{\partial x^2\partial y} \left( x_*,y,t_{n+\frac{1}{2}}\right) dxdy \\&+ \iint _{T_{i,j-\frac{1}{2}}}\frac{1}{\tau }\int _{t_n}^{t_{n+1}} \frac{1}{2}\left( t-t_{n+\frac{1}{2}}\right) ^2\frac{\partial ^3 H}{\partial t^2\partial y}(x,y,t_*)dtdydx \\= & {} \left( O\left( h_x^2\right) \left| \frac{\partial ^3 H}{\partial x^2\partial y}\right| _{\infty } + O(\tau ^2)\left| \frac{\partial ^3 H}{\partial t^2\partial y}\right| _{\infty }\right) \left| T_{i,j-\frac{1}{2}}\right| , \end{aligned}$$
where \(x_*\) is some number between \(x_{i+\frac{1}{2}}\) and x, and \(t_*\) is some number between \(t_{n+\frac{1}{2}}\) and t.
Using exactly the same argument, we can estimate \(R_{13}\) as follows:
$$\begin{aligned} R_{13}= & {} \iint _{T_{i,j-\frac{1}{2}}} \left( J_x\left( x_{i+\frac{1}{2}},y_j,t_{n+\frac{1}{2}}\right) -J_x\left( x,y,t_{n+\frac{1}{2}}\right) \right) dxdy \\&+\frac{1}{\tau }\int _{t_n}^{t_{n+1}}\iint _{T_{i,j-\frac{1}{2}}} \left( J_x\left( x,y,t_{n+\frac{1}{2}}\right) -J_x(x,y,t)\right) dxdydt \\= & {} \left( O\left( h_x^2\right) \left| \frac{\partial ^2 J_x}{\partial x^2}\right| _{\infty } + O\left( h_y^2\right) \left| \frac{\partial ^2 J_x}{\partial y^2}\right| _{\infty } + O(\tau ^2)\left| \frac{\partial ^2 J_x}{\partial t^2}\right| _{\infty }\right) \left| T_{i,j-\frac{1}{2}}\right| . \end{aligned}$$
3.2.2 The error equation for \(E_y\)
Multiplying (35) by \(|T_{i-\frac{1}{2},j}|\), we can easily derive the error equation for \(E_y\):
$$\begin{aligned} \frac{\epsilon _0\left| T_{i-\frac{1}{2},j}\right| }{\tau } \left( \mathcal {E}_{y,i,j+\frac{1}{2}}^{n+1}-\mathcal {E}_{y,i,j+\frac{1}{2}}^{n}\right)= & {} -(y_{j+1}-y_j) \left( \mathcal {H}_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}} -\mathcal {H}_{i-\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}}\right) \nonumber \\&-\left| T_{i-\frac{1}{2},j}\right| \mathcal {J}_{y,i,j+\frac{1}{2}}^{n+\frac{1}{2}}+R_2, \end{aligned}$$
(60)
where the local truncation error \(R_2\) is given by
$$\begin{aligned} R_2= & {} \frac{\epsilon _0\left| T_{i-\frac{1}{2},j}\right| }{\tau } \left( E_y\left( x_i,y_{j+\frac{1}{2}},t_{n+1}\right) - E_y\left( x_i,y_{j+\frac{1}{2}},t_{n}\right) \right) \nonumber \\&+\,(y_{j+1}-y_j)\left( H\left( x_{i+\frac{1}{2}},y_{j+\frac{1}{2}},t_{n+\frac{1}{2}}\right) -H\left( x_{i-\frac{1}{2}},y_{j+\frac{1}{2}},t_{n+\frac{1}{2}}\right) \right) \nonumber \\&+\,\left| T_{i-\frac{1}{2},j}\right| J_y\left( x_i,y_{j+\frac{1}{2}},t_{n+\frac{1}{2}}\right) . \end{aligned}$$
(61)
Integrating (9) from \(t=t_n\) to \(t_{n+1}\) and dividing the resultant by \(\tau \), we have
$$\begin{aligned}&\frac{\epsilon _0}{\tau }\iint _{T_{i-\frac{1}{2},j}} \left( E_y(x,y,t_{n+1})-E_y(x,y,t_{n})\right) dxdy \nonumber \\&\quad =-\frac{1}{\tau }\int _{t_n}^{t_{n+1}}\int _{y_j}^{y_{j+1}} \left( H\left( x_{i+\frac{1}{2}},y,t\right) -H\left( x_{i-\frac{1}{2}},y,t\right) \right) dydt\nonumber \\&\qquad -\frac{1}{\tau }\int _{t_n}^{t_{n+1}}\iint _{T_{i-\frac{1}{2}},j}J_y(x,y,t)dxdydt. \end{aligned}$$
(62)
Subtracting (62) from (61), we can rewrite \(R_2\) as follows:
$$\begin{aligned} R_2= & {} \frac{\epsilon _0}{\tau }\iint _{T_{i-\frac{1}{2},j}} \left[ \left( E_y(x_i,y_{j+\frac{1}{2}},t_{n+1}) -E_y(x,y,t_{n+1})\right) \right. \nonumber \\&\left. -\left( E_y\left( x_i,y_{j+\frac{1}{2}},t_n\right) -E_y(x,y,t_{n})\right) \right] dxdy \nonumber \\&-\left\{ \int _{y_j}^{y_{j+1}} \left( H\left( x_{i+\frac{1}{2}},y_{j+\frac{1}{2}},t_{n+\frac{1}{2}}\right) -H\left( x_{i-\frac{1}{2}},y_{j+\frac{1}{2}},t_{n+\frac{1}{2}}\right) \right) dy \right. \nonumber \\&\left. -\frac{1}{\tau }\int _{t_n}^{t_{n+1}}\int _{y_j}^{y_{j+1}} \left( H\left( x_{i+\frac{1}{2}},y,t\right) -H\left( x_{i-\frac{1}{2}},y,t\right) \right) dydt \right\} \nonumber \\&+\left[ \iint _{T_{i-\frac{1}{2}},j} J_y\left( x_i,y_{j+\frac{1}{2}},t_{n+\frac{1}{2}}\right) dxdy -\frac{1}{\tau }\int _{t_n}^{t_{n+1}} \iint _{T_{i-\frac{1}{2},j}}J_y(x,y,t)dxdydt\right] \nonumber \\= & {} R_{21} + R_{22} + R_{23}. \end{aligned}$$
(63)
Following exactly the same technique developed above for \(R_1\), we can show that
$$\begin{aligned}&R_{21}=\left( O\left( h_x^2\right) \left| \frac{\partial ^3 E_y}{\partial t\partial x^2}\right| _{\infty } + O\left( h_y^2\right) \left| \frac{\partial ^3 E_y}{\partial t\partial y^2}\right| _{\infty }\right) \left| T_{i-\frac{1}{2},j}\right| , \\&R_{22}= \left( O\left( h_y^2\right) \left| \frac{\partial ^3 H}{\partial y^2\partial x}\right| _{\infty } + O(\tau ^2)\left| \frac{\partial ^3 H}{\partial t^2\partial x}\right| _{\infty }\right) \left| T_{i-\frac{1}{2},j}\right| , \\&R_{23}=\left( O\left( h_x^2\right) \left| \frac{\partial ^2 J_y}{\partial x^2}\right| _{\infty } + O\left( h_y^2\right) \left| \frac{\partial ^2 J_y}{\partial y^2}\right| _{\infty } + O(\tau ^2)\left| \frac{\partial ^2 J_y}{\partial t^2}\right| _{\infty }\right) \left| T_{i-\frac{1}{2},j}\right| . \end{aligned}$$
3.2.3 The error equation for H
Multiplying (36) by \(|T_{i,j}|\), we can easily obtain the error equation for H:
$$\begin{aligned}&\frac{\mu _0|T_{i,j}|}{\tau } \left( \mathcal {H}_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{3}{2}} -\mathcal {H}_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}}\right) =-(y_{j+1}-y_j) \left( \mathcal {E}_{y,i+1,j+\frac{1}{2}}^{n+1}-\mathcal {E}_{y,i,j+\frac{1}{2}}^{n+1}\right) \nonumber \\&\quad +\,(x_{i+1}-x_i) \left( \mathcal {E}_{x,i+\frac{1}{2},j+1}^{n+1}-\mathcal {E}_{x,i+\frac{1}{2},j}^{n+1}\right) -|T_{i,j}|\mathcal {K}_{i+\frac{1}{2},j+\frac{1}{2}}^{n+1}+R_3, \end{aligned}$$
(64)
where the local truncation error \(R_3\) is given by
$$\begin{aligned} R_3= & {} \frac{\mu _0|T_{i,j}|}{\tau } \left( H\left( x_{i+\frac{1}{2}},y_{j+\frac{1}{2}},t_{n+\frac{3}{2}}\right) - H\left( x_{i+\frac{1}{2}},y_{j+\frac{1}{2}},t_{n+\frac{1}{2}}\right) \right) \nonumber \\&+\,(y_{j+1}-y_j)\left( E_y(x_{i+1},y_{j+\frac{1}{2}},t_{n+1}) - E_y\left( x_i,y_{j+\frac{1}{2}},t_{n+1}\right) \right) \\&-\,(x_{i+1}-x_i)\left( E_x(x_{i+\frac{1}{2}},y_{j+1},t_{n+1}) - E_x\left( x_{i+\frac{1}{2}},y_{j},t_{n+1}\right) \right) \nonumber \\&+\,|T_{i,j}|K\left( x_{i+\frac{1}{2}},y_{j+\frac{1}{2}},t_{n+1}\right) . \nonumber \end{aligned}$$
(65)
Integrating (11) from \(t=t_{n+\frac{1}{2}}\) to \(t_{n+\frac{3}{2}}\) and dividing the resultant by \(\tau \), we obtain
$$\begin{aligned}&\frac{\mu _0}{\tau }\iint _{T_{i,j}} \left( H\left( x,y,t_{n+\frac{3}{2}}\right) -H\left( x,y,t_{n+\frac{1}{2}}\right) \right) dxdy\nonumber \\&\quad =-\frac{1}{\tau } \int _{t_{n+\frac{1}{2}}}^{t_{n+\frac{3}{2}}}\iint _{T_{i,j}}\left( \frac{\partial E_y}{\partial x}-\frac{\partial E_x}{\partial y}\right) (x,y,t)dxdydt \nonumber \\&\quad \quad -\frac{1}{\tau }\int _{t_{n+\frac{1}{2}}}^{t_{n+\frac{3}{2}}} \iint _{T_{i,j}} K(x,y,t)dxdydt. \end{aligned}$$
(66)
Subtracting (66) from (65), we can rewrite \(R_3\) as follows:
$$\begin{aligned} R_3= & {} \frac{\mu _0}{\tau }\iint _{T_{i,j}} \left\{ \left( H\left( x_{i+\frac{1}{2}},y_{j+\frac{1}{2}}, t_{n+\frac{3}{2}}\right) -H\left( x,y,t_{n+\frac{3}{2}}\right) \right) \right. \nonumber \\&\left. - \left( H\left( x_{i+\frac{1}{2}},y_{j+\frac{1}{2}},t_{n+\frac{1}{2}}\right) -H\left( x,y,t_{n+\frac{1}{2}}\right) \right) \right\} dxdy \nonumber \\&+\left\{ \iint _{T_{i,j}} \left( \frac{\partial E_y}{\partial x}\left( x,y_{j+\frac{1}{2}},t_{n+1}\right) -\frac{\partial E_x}{\partial y}\left( x_{i+\frac{1}{2}},y,t_{n+1}\right) \right) dxdy \right. \nonumber \\&\left. -\frac{1}{\tau }\int _{t_{n+\frac{1}{2}}}^{t_{n+\frac{3}{2}}} \iint _{T_{i,j}} \left( \frac{\partial E_y}{\partial x}(x,y,t) -\frac{\partial E_x}{\partial y}(x,y,t)\right) dxdydt\right\} \nonumber \\&+\left\{ \iint _{T_{i,j}} K\left( x_{i+\frac{1}{2}},y_{j+\frac{1}{2}},t_{n+1}\right) dxdy -\frac{1}{\tau }\int _{t_{n+\frac{1}{2}}}^{t_{n+\frac{3}{2}}} \iint _{T_{i,j}} K(x,y,t)dxdydt\right\} \nonumber \\= & {} R_{31}+R_{32}+R_{33}. \end{aligned}$$
(67)
By the Taylor expansion, we can obtain
$$\begin{aligned} R_{31}= & {} \left( O\left( h_x^2\right) \left| \frac{\partial ^3 H}{\partial t\partial x^2}\right| _{\infty } + O\left( h_y^2\right) \left| \frac{\partial ^3 H}{\partial t\partial y^2}\right| _{\infty }\right) |T_{i,j}|, \\ R_{32}= & {} \left( O\left( h_y^2\right) \left| \frac{\partial ^3 E_y}{\partial y^2\partial x}\right| _{\infty } + O(\tau ^2)\left| \frac{\partial ^3 E_y}{\partial t^2\partial x}\right| _{\infty }\right. \\&\left. +\, O\left( h_x^2\right) \left| \frac{\partial ^3 E_x}{\partial x^2\partial y}\right| _{\infty } + O(\tau ^2)\left| \frac{\partial ^3 E_x}{\partial t^2\partial y}\right| _{\infty }\right) |T_{i,j}|, \\ R_{33}= & {} \left( O\left( h_x^2\right) \left| \frac{\partial ^2 K}{\partial x^2}\right| _{\infty } + O\left( h_y^2\right) \left| \frac{\partial ^2 K}{\partial y^2}\right| _{\infty } + O(\tau ^2)\left| \frac{\partial ^2 K}{\partial t^2}\right| _{\infty }\right) |T_{i,j}|. \end{aligned}$$
3.2.4 The error equation for \(J_x\)
Multiplying (37) by \(|T_{i,j-\frac{1}{2}}|\), we easily derive the error equation for \(J_x\):
$$\begin{aligned}&\frac{\left| T_{i,j-\frac{1}{2}}\right| }{\tau \epsilon _0\omega _{pe}^2} \left( \mathcal {J}_{x,i+\frac{1}{2},j}^{n+\frac{3}{2}} -\mathcal {J}_{x,i+\frac{1}{2},j}^{n+\frac{1}{2}}\right) + \frac{\Gamma _e\left| T_{i,j-\frac{1}{2}}\right| }{2\epsilon _0\omega _{pe}^2}\left( \mathcal {J}_{x,i+\frac{1}{2},j}^{n+\frac{3}{2}} +\mathcal {J}_{x,i+\frac{1}{2},j}^{n+\frac{1}{2}}\right) \nonumber \\&\quad =\left| T_{i,j-\frac{1}{2}}\right| \mathcal {E}_{x,i+\frac{1}{2},j}^{n+1} + R_4, \end{aligned}$$
(68)
where the local truncation error \(R_4\) is given by
$$\begin{aligned} R_4= & {} \frac{\left| T_{i,j-\frac{1}{2}}\right| }{\tau \epsilon _0\omega _{pe}^2}\left( J_x\left( x_{i+\frac{1}{2}},y_j, t_{n+\frac{3}{2}}\right) -J_x\left( x_{i+\frac{1}{2}},y_j,t_{n+\frac{1}{2}}\right) \right) \nonumber \\&+\frac{\Gamma _e\left| T_{i,j-\frac{1}{2}}\right| }{2\epsilon _0\omega _{pe}^2}\left( J_x\left( x_{i+\frac{1}{2}},y_j, t_{n+\frac{3}{2}}\right) +J_x\left( x_{i+\frac{1}{2}},y_j,t_{n+\frac{1}{2}}\right) \right) \nonumber \\&- \left| T_{i,j-\frac{1}{2}}\right| E_x\left( x_{i+\frac{1}{2}},y_j,t_{n+1}\right) . \end{aligned}$$
(69)
Integrating (13) from \(t=t_{n+\frac{1}{2}}\) to \(t_{n+\frac{3}{2}}\) and dividing the resultant by \(\tau \), we have
$$\begin{aligned}&\frac{1}{\tau \epsilon _0\omega _{pe}^2} \iint _{T_{i,j-\frac{1}{2}}} \left( J_x\left( x,y,t_{n+\frac{3}{2}}\right) -J_x\left( x,y,t_{n+\frac{1}{2}}\right) \right) dxdy \nonumber \\&\qquad +\frac{\Gamma _e}{\tau \epsilon _0\omega _{pe}^2} \int _{t_{n+\frac{1}{2}}}^{t_{n+\frac{3}{2}}}\iint _{T_{i,j-\frac{1}{2}}} J_x(x,y,t)dxdydt\nonumber \\&\quad =\frac{1}{\tau }\int _{t_{n+\frac{1}{2}}}^{t_{n+\frac{3}{2}}}\iint _{T_{i,j-\frac{1}{2}}} E_x(x,y,t)dxdydt. \end{aligned}$$
(70)
Subtracting (70) from (69), we can rewrite \(R_4\) as follows:
$$\begin{aligned} R_4= & {} \frac{1}{\tau \epsilon _0\omega _{pe}^2} \iint _{T_{i,j-\frac{1}{2}}} \left\{ \left( J_x\left( x_{i+\frac{1}{2}},y_j,t_{n+\frac{3}{2}}\right) -J_x\left( x,y,t_{n+\frac{3}{2}}\right) \right) \right. \nonumber \\&\left. -\left( J_x\left( x_{i+\frac{1}{2}},y_j,t_{n+\frac{1}{2}}\right) -J_x\left( x,y,t_{n+\frac{1}{2}}\right) \right) \right\} dxdy \nonumber \\&+\frac{\Gamma _e}{\epsilon _0\omega _{pe}^2} \left\{ \iint _{T_{i,j-\frac{1}{2}}} \frac{1}{2}\left( J_x\left( x_{i+\frac{1}{2}},y_j,t_{n+\frac{3}{2}}\right) +J_x\left( x_{i+\frac{1}{2}},y_j,t_{n+\frac{1}{2}}\right) \right) dxdy \right. \nonumber \\&\left. -\frac{1}{\tau }\int _{t_{n+\frac{1}{2}}}^{t_{n+\frac{3}{2}}}\iint _{T_{i,j-\frac{1}{2}}} J_x(x,y,t)dxdydt\right\} \nonumber \\&-\left\{ \iint _{T_{i,j-\frac{1}{2}}}E_x\left( x_{i+\frac{1}{2}},y_j,t_{n+1}\right) dxdy - \frac{1}{\tau }\int _{t_{n+\frac{1}{2}}}^{t_{n+\frac{3}{2}}}\iint _{T_{i,j-\frac{1}{2}}} E_x(x,y,t)dxdydt\right\} \nonumber \\= & {} R_{41}+R_{42}+R_{43}. \end{aligned}$$
(71)
By the Taylor expansion, we easily have
$$\begin{aligned} R_{41}= & {} \left( O\left( h_x^2\right) \left| \frac{\partial ^3 J_x}{\partial t\partial x^2}\right| _{\infty } + O\left( h_y^2\right) \left| \frac{\partial ^3 J_x}{\partial t\partial y^2}\right| _{\infty }\right) \left| T_{i,j-\frac{1}{2}}\right| , \\ R_{42}= & {} \frac{\Gamma _e}{\epsilon _0\omega _{pe}^2} \bigg \{\iint _{T_{i,j-\frac{1}{2}}} \frac{1}{2}\left( J_x\left( x_{i+\frac{1}{2}},y_j,t_{n+\frac{3}{2}}\right) +J_x\left( x_{i+\frac{1}{2}},y_j,t_{n+\frac{1}{2}}\right) \right. \nonumber \\&\left. -\,J_x\left( x,y,t_{n+\frac{3}{2}}\right) -J_x\left( x,y,t_{n+\frac{1}{2}}\right) \right) dxdy\bigg \} \nonumber \\&+\,\iint _{T_{i,j-\frac{1}{2}}} \frac{1}{\tau }\int _{t_{n+\frac{1}{2}}}^{t_{n+\frac{3}{2}}} \left\{ \frac{1}{2}\left( J_x\left( x,y,t_{n+\frac{3}{2}}\right) +J_x\left( x,y,t_{n+\frac{1}{2}}\right) \right) -J_x(x,y,t)\right\} dtdxdy \nonumber \\= & {} \left( O\left( h_x^2\right) \left| \frac{\partial ^2 J_x}{\partial x^2}\right| _{\infty } + O\left( h_y^2\right) \left| \frac{\partial ^2 J_x}{\partial y^2}\right| _{\infty } + O(\tau ^2)\left| \frac{\partial ^2 J_x}{\partial t^2}\right| _{\infty }\right) \left| T_{i,j-\frac{1}{2}}\right| , \end{aligned}$$
where in the last step we used the property: For any function \(f\in C^2([0,T])\),
$$\begin{aligned} \frac{1}{\tau }\int _{t_{n+\frac{1}{2}}}^{t_{n+\frac{3}{2}}} \left\{ \frac{1}{2}\left( f\left( t_{n+\frac{3}{2}}\right) +f\left( t_{n+\frac{1}{2}}\right) \right) -f(t)\right\} dt=O(\tau ^2)\left| \frac{\partial ^2 f}{\partial t^2}\right| _{\infty }. \end{aligned}$$
Similarly, it is easy to show that
$$\begin{aligned} R_{43}=\left( O\left( h_x^2\right) \left| \frac{\partial ^2 E_x}{\partial x^2}\right| _{\infty } + O\left( h_y^2\right) \left| \frac{\partial ^2 E_x}{\partial y^2}\right| _{\infty } + O(\tau ^2)\left| \frac{\partial ^2 E_x}{\partial t^2}\right| _{\infty }\right) \left| T_{i,j-\frac{1}{2}}\right| . \end{aligned}$$
3.2.5 The error equation for \(J_y\)
Following exactly the same technique used for the \(J_x\) equation, we easily obtain the error equation for \(J_y\) from (38):
$$\begin{aligned}&\frac{\left| T_{i-\frac{1}{2},j}\right| }{\tau \epsilon _0\omega _{pe}^2} \left( \mathcal {J}_{y,i,j+\frac{1}{2}}^{n+\frac{3}{2}} -\mathcal {J}_{y,i,j+\frac{1}{2}}^{n+\frac{1}{2}}\right) + \frac{\Gamma _e\left| T_{i-\frac{1}{2},j}\right| }{2\epsilon _0\omega _{pe}^2}\left( \mathcal {J}_{y,i,j +\frac{1}{2}}^{n+\frac{3}{2}} +\mathcal {J}_{y,i,j +\frac{1}{2}}^{n+\frac{1}{2}}\right) \nonumber \\&\quad =\left| T_{i-\frac{1}{2},j}\right| \mathcal {E}_{y,i,j+\frac{1}{2}}^{n+1} + R_5, \end{aligned}$$
(72)
where the local truncation error \(R_5\) is given by
$$\begin{aligned} R_5= & {} \frac{\left| T_{i-\frac{1}{2},j}\right| }{\tau \epsilon _0\omega _{pe}^2}\left( J_y\left( x_i,y_{j+\frac{1}{2}}, t_{n+\frac{3}{2}}\right) -J_y\left( x_i,y_{j+\frac{1}{2}}, t_{n+\frac{1}{2}}\right) \right) \nonumber \\&+\,\frac{\Gamma _e\left| T_{i-\frac{1}{2},j}\right| }{2\epsilon _0\omega _{pe}^2}\left( J_y\left( x_i,y_{j+\frac{1}{2}}, t_{n+\frac{3}{2}}\right) +J_y\left( x_i,y_{j+\frac{1}{2}},t_{n+\frac{1}{2}}\right) \right) \nonumber \\&-\, \left| T_{i-\frac{1}{2},j} \right| E_y\left( x_i,y_{j+\frac{1}{2}},t_{n+1}\right) . \end{aligned}$$
(73)
Integrating the y-component of (3) on \(T_{i-\frac{1}{2},j}\), then integrating the resultant from \(t=t_{n+\frac{1}{2}}\) to \(t_{n+\frac{3}{2}}\) and dividing the resultant by \(\tau \), we have
$$\begin{aligned}&\frac{1}{\tau \epsilon _0\omega _{pe}^2}\iint _{T_{i-\frac{1}{2},j}} \left( J_y\left( x,y,t_{n+\frac{3}{2}}\right) -J_y\left( x,y,t_{n+\frac{1}{2}}\right) \right) dxdy \nonumber \\&\qquad +\frac{\Gamma _e}{\tau \epsilon _0\omega _{pe}^2} \int _{t_{n+\frac{1}{2}}}^{t_{n+\frac{3}{2}}}\iint _{T_{i-\frac{1}{2},j}} J_y(x,y,t)dxdydt\nonumber \\&\quad =\frac{1}{\tau }\int _{t_{n+\frac{1}{2}}}^{t_{n+\frac{3}{2}}}\iint _{T_{i-\frac{1}{2},j}} E_y(x,y,t)dxdydt. \end{aligned}$$
(74)
Subtracting (74) from (73), we can rewrite \(R_5\) as follows:
$$\begin{aligned} R_5= & {} \frac{1}{\tau \epsilon _0\omega _{pe}^2} \iint _{T_{i-\frac{1}{2},j}} \left\{ \left( J_y\left( x_i,y_{j+\frac{1}{2}},t_{n+\frac{3}{2}}\right) -J_y\left( x,y,t_{n+\frac{3}{2}}\right) \right) \right. \nonumber \\&\left. -\left( J_y\left( x_i,y_{j+\frac{1}{2}}, t_{n+\frac{1}{2}}\right) -J_y\left( x,y,t_{n+\frac{1}{2}}\right) \right) \right\} dxdy \nonumber \\&+\frac{\Gamma _e}{\epsilon _0\omega _{pe}^2} \left\{ \iint _{T_{i-\frac{1}{2},j}} \frac{1}{2}\left( J_y\left( x_i,y_{j+\frac{1}{2}},t_{n+\frac{3}{2}}\right) +J_y\left( x_i,y_{j+\frac{1}{2}},t_{n+\frac{1}{2}}\right) \right) dxdy \right. \nonumber \\&\left. -\frac{1}{\tau }\int _{t_{n+\frac{1}{2}}}^{t_{n+\frac{3}{2}}}\iint _{T_{i-\frac{1}{2},j}} J_y(x,y,t)dxdydt\right\} \nonumber \\&-\left\{ \iint _{T_{i-\frac{1}{2},j}} E_y\left( x_i,y_{j+\frac{1}{2}},t_{n+1}\right) dxdy - \frac{1}{\tau }\int _{t_{n+\frac{1}{2}}}^{t_{n+\frac{3}{2}}} \iint _{T_{i-\frac{1}{2},j}} E_y(x,y,t)dxdydt\right\} \nonumber \\= & {} R_{51}+R_{52}+R_{53}. \end{aligned}$$
(75)
By the Taylor expansion, we can obtain
$$\begin{aligned}&R_{51}=\left( O\left( h_x^2\right) \left| \frac{\partial ^3 J_y}{\partial t\partial x^2}\right| _{\infty } + O\left( h_y^2\right) \left| \frac{\partial ^3 J_y}{\partial t\partial y^2}\right| _{\infty }\right) \left| T_{i-\frac{1}{2},j}\right| , \\&R_{52}= \left( O\left( h_x^2\right) \left| \frac{\partial ^2 J_y}{\partial x^2}\right| _{\infty } + O\left( h_y^2\right) \left| \frac{\partial ^2 J_y}{\partial y^2}\right| _{\infty } + O(\tau ^2)\left| \frac{\partial ^2 J_y}{\partial t^2}\right| _{\infty }\right) \left| T_{i-\frac{1}{2},j}\right| , \\&R_{53}=\left( O\left( h_x^2\right) \left| \frac{\partial ^2 E_y}{\partial x^2}\right| _{\infty } + O\left( h_y^2\right) \left| \frac{\partial ^2 E_y}{\partial y^2}\right| _{\infty } + O(\tau ^2)\left| \frac{\partial ^2 E_y}{\partial t^2}\right| _{\infty }\right) \left| T_{i-\frac{1}{2},j}\right| . \end{aligned}$$
3.2.6 The error equation for K
Similarly, we can obtain the error equation for K from (39):
$$\begin{aligned}&\frac{|T_{i,j}|}{\tau \mu _0\omega _{pm}^2} \left( \mathcal {K}_{i+\frac{1}{2},j+\frac{1}{2}}^{n+2} -\mathcal {K}_{i+\frac{1}{2},j+\frac{1}{2}}^{n+1}\right) + \frac{\Gamma _m|T_{i,j}|}{2\mu _0\omega _{pm}^2} \left( \mathcal {K}_{i+\frac{1}{2},j+\frac{1}{2}}^{n+2} +\mathcal {K}_{i+\frac{1}{2},j+\frac{1}{2}}^{n+1}\right) \nonumber \\&\quad =|T_{i,j}|\mathcal {H}_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{3}{2}} + R_6, \end{aligned}$$
(76)
where the local truncation error \(R_6\) is given by
$$\begin{aligned} R_6= & {} \frac{|T_{i,j}|}{\tau \mu _0\omega _{pm}^2} \left( K\left( x_{i+\frac{1}{2}},y_{j+\frac{1}{2}}, t_{n+2}\right) -K\left( x_{i+\frac{1}{2}},y_{j+\frac{1}{2}},t_{n+1}\right) \right) \nonumber \\&+\frac{\Gamma _m|T_{i,j}|}{2\mu _0\omega _{pm}^2} \left( K\left( x_{i+\frac{1}{2}},y_{j+\frac{1}{2}}, t_{n+2}\right) +K\left( x_{i+\frac{1}{2}},y_{j+\frac{1}{2}},t_{n+1}\right) \right) \nonumber \\&- |T_{i,j}|H\left( x_{i+\frac{1}{2}},y_{j+\frac{1}{2}},t_{n+\frac{3}{2}}\right) . \end{aligned}$$
(77)
Integrating (4) on \(T_{i,j}\), then integrating the resultant from \(t=t_{n+1}\) to \(t_{n+2}\) and dividing the resultant by \(\tau \), we have
$$\begin{aligned}&\frac{1}{\tau \mu _0\omega _{pm}^2}\iint _{T_{i,j}} (K(x,y,t_{n+2})-K(x,y,t_{n+1}))dxdy \nonumber \\&\quad +\frac{\Gamma _m}{\tau \mu _0\omega _{pm}^2} \int _{t_{n+1}}^{t_{n+2}}\iint _{T_{i,j}}K(x,y,t)dxdydt =\frac{1}{\tau }\int _{t_{n+1}}^{t_{n+2}}\iint _{T_{i,j}} H(x,y,t)dxdydt. \nonumber \\ \end{aligned}$$
(78)
Subtracting (78) from (77), we can rewrite \(R_6\) as follows:
$$\begin{aligned} R_6= & {} \frac{1}{\tau \mu _0\omega _{pm}^2}\iint _{T_{i,j}} \left\{ \left( K\left( x_{i+\frac{1}{2}},y_{j+\frac{1}{2}},t_{n+2}\right) -K\left( x,y,t_{n+2}\right) \right) \right. \nonumber \\&\left. -\left( K\left( x_{i+\frac{1}{2}},y_{j+\frac{1}{2}},t_{n+1}\right) -K(x,y,t_{n+1})\right) \right\} dxdy \nonumber \\&+\frac{\Gamma _m}{\mu _0\omega _{pm}^2} \left\{ \iint _{T_{i,j}} \frac{1}{2}\left( K\left( x_{i+\frac{1}{2}},y_{j+\frac{1}{2}},t_{n+2}\right) +K\left( x_{i+\frac{1}{2}},y_{j+\frac{1}{2}},t_{n+1}\right) \right) dxdy \right. \nonumber \\&\left. -\frac{1}{\tau }\int _{t_{n+1}}^{t_{n+2}}\iint _{T_{i,j}} K(x,y,t)dxdydt\right\} \nonumber \\&-\left\{ \iint _{T_{i,j}}H\left( x_{i+\frac{1}{2}},y_{j+\frac{1}{2}},t_{n+\frac{3}{2}}\right) dxdy - \frac{1}{\tau }\int _{t_{n+1}}^{t_{n+2}}\iint _{T_{i,j}} H(x,y,t)dxdydt\right\} \nonumber \\= & {} R_{61}+R_{62}+R_{63}. \end{aligned}$$
(79)
By the Taylor expansion, we can obtain
$$\begin{aligned}&R_{61}=\left( O\left( h_x^2\right) \left| \frac{\partial ^3 K}{\partial t\partial x^2}\right| _{\infty } + O\left( h_y^2\right) \left| \frac{\partial ^3 K}{\partial t\partial y^2}\right| _{\infty }\right) |T_{i,j}|, \\&R_{62}= \left( O\left( h_x^2\right) \left| \frac{\partial ^2 K}{\partial x^2}\right| _{\infty } + O\left( h_y^2\right) \left| \frac{\partial ^2 K}{\partial y^2}\right| _{\infty } + O(\tau ^2)\left| \frac{\partial ^2 K}{\partial t^2}\right| _{\infty }\right) |T_{i,j}|, \\&R_{63}=\left( O\left( h_x^2\right) \left| \frac{\partial ^2 H}{\partial x^2}\right| _{\infty } + O\left( h_y^2\right) \left| \frac{\partial ^2 H}{\partial y^2}\right| _{\infty } + O(\tau ^2)\left| \frac{\partial ^2 H}{\partial t^2}\right| _{\infty }\right) |T_{i,j}|. \end{aligned}$$
3.2.7 The final error estimate
With the above preparations, we can now prove the major error estimate result.
Theorem 3.2
Suppose that the solution of (1)–(6) possesses the following regularity property:
$$\begin{aligned}&E_x, E_y, H \in C([0,T]; C^3(\overline{\Omega }))\cap C^1([0,T]; C^2(\overline{\Omega }))\cap C^2([0,T]; C^1(\overline{\Omega })), \\&J_x, J_y, K \in C([0,T]; C^2(\overline{\Omega }))\cap C^1([0,T]; C^2(\overline{\Omega }))\cap C^2([0,T]; C(\overline{\Omega })). \end{aligned}$$
If the initial error
$$\begin{aligned} \left| \left| \mathcal {E}_x^{0}\right| \right| _E +\left| \left| \mathcal {E}_y^{0}\right| \right| _E +\left| \left| \mathcal {H}^{\frac{1}{2}}\right| \right| _H +\left| \left| \mathcal {J}_x^{\frac{1}{2}}\right| \right| _J +\left| \left| \mathcal {J}_y^{\frac{1}{2}}\right| \right| _J +\left| \left| \mathcal {K}^{1}\right| \right| _K \le C\left( h_x^2+h_y^2+\tau ^2\right) , \end{aligned}$$
(80)
holds true, then for any \(1\le n\le N_t\) we have
$$\begin{aligned}&\epsilon _0\left( \left| \left| \mathcal {E}_x^{n+1}\right| \right| _E^2 +\left| \left| \mathcal {E}_y^{n+1}\right| \right| _E^2\right) +\mu _0\left| \left| \mathcal {H}^{n+\frac{3}{2}}\right| \right| ^2_H\nonumber \\&\qquad +\frac{1}{\epsilon _0\omega _{pe}^2} \left( \left| \left| \mathcal {J}_x^{n+\frac{3}{2}}\right| \right| ^2_J +\left| \left| \mathcal {J}_y^{n+\frac{3}{2}}\right| \right| ^2_J\right) +\frac{1}{\mu _0\omega _{pm}^2}\left| \left| \mathcal {K}^{n+2}\right| \right| ^2_K \nonumber \\&\quad \le C\left( h_x^2+h_y^2+\tau ^2\right) ^2, \end{aligned}$$
(81)
where the constant \(C>0\) is independent of \(\tau , h_x\) and \(h_y.\)
Proof
Note that the error equations (55), (60), (64), (68), (72) and (76) have exactly the same form as (34)–(39) with extra right hand side terms representing the errors introduced by time discretization and space discretization. Hence we can follow exactly the same technique developed in the proof of Theorem 3.1 to obtain (cf. (49)):
$$\begin{aligned}&\epsilon _0\left( \left| \left| \mathcal {E}_x^{N_t+1}\right| \right| ^2_E -\left| \left| \mathcal {E}_x^{0}\right| \right| ^2_E\right) +\epsilon _0\left( \left| \left| \mathcal {E}_y^{N_t+1}\right| \right| ^2_E -\left| \left| \mathcal {E}_y^{0}\right| \right| ^2_E\right) \nonumber \\&\qquad +\,\mu _0\left( \left| \left| \mathcal {H}^{N_t+\frac{3}{2}}\right| \right| ^2_H -\left| \left| \mathcal {H}^{\frac{1}{2}}\right| \right| ^2_H\right) \nonumber \\&\qquad + \frac{1}{\epsilon _0\omega ^2_{pe}} \left( \left| \left| \mathcal {J}_x^{N_t+\frac{3}{2}}\right| \right| ^2_J -\left| \left| \mathcal {J}_x^{\frac{1}{2}}\right| \right| ^2_J\right) +\frac{1}{\epsilon _0\omega ^2_{pe}} \left( \left| \left| \mathcal {J}_y^{N_t+\frac{3}{2}}\right| \right| ^2_J -\left| \left| \mathcal {J}_y^{\frac{1}{2}}\right| \right| ^2_J\right) \nonumber \\ \end{aligned}$$
$$\begin{aligned}&\qquad + \frac{1}{\mu _0\omega ^2_{pm}}\left( \left| \left| \mathcal {K}^{N_t+2}\right| \right| ^2_K -\left| \left| \mathcal {K}^{1}\right| \right| ^2_K\right) \nonumber \\&\quad \le \tau \sum _{\begin{array}{c} 0\le i \le N_x-1 \\ 0\le j \le N_y-1 \end{array}}|T_{ij}| \left( \mathcal {H}_{i+\frac{1}{2},j+\frac{1}{2}}^{N_t+\frac{3}{2}} \nabla _y\mathcal {E}_{x,i+\frac{1}{2},j+1}^{N_t+1} -\mathcal {H}_{i+\frac{1}{2},j+\frac{1}{2}}^{\frac{1}{2}} \nabla _y\mathcal {E}_{x,i+\frac{1}{2},j+1}^{0}\right) \nonumber \\&\qquad +\,\tau \sum _{\begin{array}{c} 0\le i \le N_x-1 0\le j \le N_y-1 \end{array}}|T_{ij}| \left( -\mathcal {H}_{i+\frac{1}{2},j+\frac{1}{2}}^{N_t+\frac{3}{2}} \nabla _x\mathcal {E}_{y,i+1,j+\frac{1}{2}}^{N_t+1} +\mathcal {H}_{i+\frac{1}{2},j+\frac{1}{2}}^{\frac{1}{2}}\nabla _x\mathcal {E}_{y,i+1,j+\frac{1}{2}}^{0}\right) \nonumber \\ \end{aligned}$$
$$\begin{aligned}&\qquad +\, \tau \sum _{\begin{array}{c} 0\le i \le N_x-1 0\le j \le N_y-1 \end{array}}\left| T_{i,j-\frac{1}{2}}\right| \left( \mathcal {J}_{x,i+\frac{1}{2},j}^{N_t+\frac{3}{2}}\mathcal {E}_{x,i+\frac{1}{2},j}^{N_t+1} -\mathcal {J}_{x,i+\frac{1}{2},j}^{\frac{1}{2}}\mathcal {E}_{x,i+\frac{1}{2},j}^{0}\right) \nonumber \\&\qquad +\,\tau \sum _{\begin{array}{c} 0\le i \le N_x-1 0\le j \le N_y-1 \end{array}}\left| T_{i-\frac{1}{2},j}\right| \left( \mathcal {J}_{y,i,j+\frac{1}{2}}^{N_t+\frac{3}{2}}\mathcal {E}_{y,i,j+\frac{1}{2}}^{N_t+1} -\mathcal {J}_{y,i,j+\frac{1}{2}}^{\frac{1}{2}}\mathcal {E}_{y,i,j+\frac{1}{2}}^{0}\right) \nonumber \\&\qquad +\,\tau \sum _{\begin{array}{c} 0\le i \le N_x-1 0\le j \le N_y-1 \end{array}}|T_{ij}| \left( \mathcal {H}_{i+\frac{1}{2},j+\frac{1}{2}}^{N_t+\frac{3}{2}}\mathcal {K}_{i+\frac{1}{2},j+\frac{1}{2}}^{N_t+2} -\mathcal {H}_{i+\frac{1}{2},j+\frac{1}{2}}^{\frac{1}{2}}\mathcal {K}_{i+\frac{1}{2},j+\frac{1}{2}}^{1}\right) \nonumber \\&\qquad +\, \tau \sum _{n=0}^{N_t}\sum _{\begin{array}{c} 0\le i \le N_x-1 0\le j \le N_y-1 \end{array}} R_1\left( \mathcal {E}_{x,i+\frac{1}{2},j}^{n+1}+\mathcal {E}_{x,i+\frac{1}{2},j}^{n}\right) \nonumber \\&\qquad +\,\tau \sum _{n=0}^{N_t}\sum _{\begin{array}{c} 0\le i \le N_x-1 0\le j \le N_y-1 \end{array}} R_2\left( \mathcal {E}_{y,i,j+\frac{1}{2}}^{n+1}+\mathcal {E}_{y,i,j+\frac{1}{2}}^{n}\right) \nonumber \\ \end{aligned}$$
$$\begin{aligned}&\qquad +\, \tau \sum _{n=0}^{N_t}\sum _{\begin{array}{c} 0\le i \le N_x-1 0\le j \le N_y-1 \end{array}} R_3\left( \mathcal {H}_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{3}{2}}+\mathcal {H}_{i+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}}\right) \nonumber \\&\qquad +\,\tau \sum _{n=0}^{N_t}\sum _{\begin{array}{c} 0\le i \le N_x-1 0\le j \le N_y-1 \end{array}} R_4\left( \mathcal {J}_{x,i+\frac{1}{2},j}^{n+\frac{3}{2}}+\mathcal {J}_{x,i+\frac{1}{2},j}^{n+\frac{1}{2}}\right) \nonumber \\&\qquad +\,\tau \sum _{n=0}^{N_t}\sum _{\begin{array}{c} 0\le i \le N_x-1 0\le j \le N_y-1 \end{array}} R_5\left( \mathcal {J}_{y,i,j+\frac{1}{2}}^{n+\frac{3}{2}}+\mathcal {J}_{y,i,j+\frac{1}{2}}^{n+\frac{1}{2}}\right) \nonumber \\&\qquad +\,\tau \sum _{n=0}^{N_t}\sum _{\begin{array}{c} 0\le i \le N_x-1 0\le j \le N_y-1 \end{array}} R_6\left( \mathcal {K}_{i+\frac{1}{2},j+\frac{1}{2}}^{n+2}+\mathcal {K}_{i+\frac{1}{2},j+\frac{1}{2}}^{n+1}\right) . \end{aligned}$$
(82)
All terms except those containing \(R_i\) on the RHS of (82) can be bounded as in the proof of Theorem 3.1. The \(R_i\) terms can be easily bounded by the Cauchy–Schwarz inequality. For example, we have
$$\begin{aligned}&\tau \sum _{n=0}^{N_t}\sum _{\begin{array}{c} 0\le i \le N_x-1 \\ 0\le j \le N_y-1 \end{array}} R_1\left( \mathcal {E}_{x,i+\frac{1}{2},j}^{n+1}+\mathcal {E}_{x,i+\frac{1}{2},j}^{n}\right) \\&\quad \le \tau \sum _{n=0}^{N_t}\sum _{\begin{array}{c} 0\le i \le N_x-1 \\ 0\le j \le N_y-1 \end{array}} \left| T_{i,j-\frac{1}{2}}\right| C\left( h_x^2+h_y^2+\tau ^2\right) \left( \left| \mathcal {E}_{x,i+\frac{1}{2},j}^{n+1} +\mathcal {E}_{x,i+\frac{1}{2},j}^{n}\right| \right) \\&\quad \le \tau \sum _{n=0}^{N_t}\sum _{\begin{array}{c} 0\le i \le N_x-1 \\ 0\le j \le N_y-1 \end{array}} \left| T_{i,j-\frac{1}{2}}\right| \left[ \frac{C}{\delta } \left( h_x^2+h_y^2+\tau ^2\right) ^2 +\frac{\delta }{2} \left( \left| \mathcal {E}_{x,i+\frac{1}{2},j}^{n+1}|^2 +|\mathcal {E}_{x,i+\frac{1}{2},j}^{n}\right| ^2\right) \right] \\&\quad \le \frac{CT}{\delta }\left( h_x^2+h_y^2+\tau ^2\right) ^2 +\tau \sum _{n=0}^{N_t}\frac{\delta }{2} \left( \left| \left| \mathcal {E}_{x,i+\frac{1}{2},j}^{n+1}\right| \right| _E^2 +\left| \left| \mathcal {E}_{x,i+\frac{1}{2},j}^{n}\right| \right| _E^2\right) , \end{aligned}$$
where we used the inequality \(ab\le \frac{1}{\delta }a^2 + \frac{\delta }{4}b^2\), where the constant \(\delta >0\).
Choosing \(\delta \) small enough so that \(||\mathcal {E}_{x,i+\frac{1}{2},j}^{N_t+1}||_E^2\) etc can be bounded by the corresponding terms on the left hand side of (82). The proof is completed by using the discrete Gronwall inequality. \(\square \)