## 5 Discussion and Summary
ADE: If we analyze the core Formulas (9) and (10), where only nearest neighbors are taken into account, we obtain: \[\delta_{\text{UPFD}}=-\frac{\beta^{2}}{2}\big{(}u_{i\pm 1}^{n}-2u_{i}^{n} \big{)}\Delta t+\frac{5\beta^{3}}{2}\big{(}u_{i\pm 1}^{n}-2u_{i}^{n}\big{)} \Delta t^{2}+O\Big{(}\Delta t^{3}\Big{)}.\]5 \[\delta_{\text{UPFD}}=-\frac{\beta^{2}}{2}\big{(}u_{i\pm 1}^{n}-2u_{i}^{n} \big{)}\Delta t+\frac{5\beta^{3}}{2}\big{(}u_{i\pm 1}^{n}-2u_{i}^{n}\big{)} \Delta t^{2}+O\Big{(}\Delta t^{3}\Big{)}.\]6  
The expression of \(\tau\) and \(\varepsilon\) are definitely not the same. The first one demonstrates a second-order consistent scheme; the second is first-order and conditionally consistent. According to the numerical experiments, ADE is second-order but conditionally consistent. The problem may be caused by the fact that (9) and (10) are actually implicit formulas and only the alternating direction splitting procedure makes them explicit. In other words, the information where the \(p_{i-1}^{n+1}\) and \(q_{i+1}^{n+1}\) values come from is not present in (9) and (10). Now we make an attempt to involve the whole alternating direction procedure to obtain a more reliable result. This is performed by expressing the new values \(p_{i-1}^{n+1}\) and \(q_{i+1}^{n+1}\) of the neighbors by formulas (10) with indices shifted by one, and substituted into (9) and (10) again. With this, one extra neighbor is involved on both sides and we obtain:  
\[\delta_{\text{UPFD}}=-\frac{\beta^{2}}{2}\big{(}u_{i\pm 1}^{n}-2u_{i}^{n} \big{)}\Delta t+\frac{5\beta^{3}}{2}\big{(}u_{i\pm 1}^{n}-2u_{i}^{n}\big{)} \Delta t^{2}+O\Big{(}\Delta t^{3}\Big{)}.\]7  
\[\delta_{\text{UPFD}}=-\frac{\beta^{2}}{2}\big{(}u_{i\pm 1}^{n}-2u_{i}^{n} \big{)}\Delta t+\frac{5\beta^{3}}{2}\big{(}u_{i\pm 1}^{n}-2u_{i}^{n}\big{)} \Delta t^{2}+O\Big{(}\Delta t^{3}\Big{)}.\]8  
Repeating this procedure recursively, two or more extra neighbors are involved. In these cases we obtain:  
\[\delta_{\text{UPFD}}=-\frac{\beta^{2}}{2}\big{(}u_{i\pm 1}^{n}-2u_{i}^{n} \big{)}\Delta t+\frac{5\beta^{3}}{2}\big{(}u_{i\pm 1}^{n}-2u_{i}^{n}\big{)} \Delta t^{2}+O\Big{(}\Delta t^{3}\Big{)}.\]9  
\[\tau_{\text{CNe}}=\varepsilon_{\text{CNe}}=\varepsilon_{0}+\alpha^{2}\frac{u _{(2x)}}{\Delta x^{2}}\Delta t+\frac{7}{12}\alpha^{2}u_{(4x)}\Delta t+\frac{ \alpha^{2}}{360}u_{(6x)}\Delta x^{2}\Delta t-\frac{2}{3}\alpha^{3}u_{(2x)} \frac{\Delta t^{2}}{\Delta x^{4}}-\frac{\alpha^{3}}{18}u_{(4x)}\frac{\Delta t ^{2}}{\Delta x^{2}}+\frac{89}{540}\alpha^{3}u_{(6x)}\Delta t^{2}\]0  
We believe that this last expression reflects the real nature of the numerical method since it means a second-order scheme, which is conditionally consistent. However, the leading inconsistent term contains \(\Delta t^{2}\Delta x^{-2}\), instead of \(\Delta t^{2}\Delta x^{-4}\) as in the case of the previous methods, thus this method converges much faster than those. 11. DF: We have to face a similar problem as in the case of the ADE method. If the core formulas are the starting point, we obtain different expressions for \(\tau\) and \(\varepsilon\): \[\tau_{\text{CNe}}=\varepsilon_{\text{CNe}}=\varepsilon_{0}+\alpha^{2}\frac{u _{(2x)}}{\Delta x^{2}}\Delta t+\frac{7}{12}\alpha^{2}u_{(4x)}\Delta t+\frac{ \alpha^{2}}{360}u_{(6x)}\Delta x^{2}\Delta t-\frac{2}{3}\alpha^{3}u_{(2x)} \frac{\Delta t^{2}}{\Delta x^{4}}-\frac{\alpha^{3}}{18}u_{(4x)}\frac{\Delta t ^{2}}{\Delta x^{2}}+\frac{89}{540}\alpha^{3}u_{(6x)}\Delta t^{2}\]1 \[\tau_{\text{CNe}}=\varepsilon_{\text{CNe}}=\varepsilon_{0}+\alpha^{2}\frac{u _{(2x)}}{\Delta x^{2}}\Delta t+\frac{7}{12}\alpha^{2}u_{(4x)}\Delta t+\frac{ \alpha^{2}}{360}u_{(6x)}\Delta x^{2}\Delta t-\frac{2}{3}\alpha^{3}u_{(2x)} \frac{\Delta t^{2}}{\Delta x^{4}}-\frac{\alpha^{3}}{18}u_{(4x)}\frac{\Delta t ^{2}}{\Delta x^{2}}+\frac{89}{540}\alpha^{3}u_{(6x)}\Delta t^{2}\]2 We note that \(\tau_{\text{DF}}\) contains the terms which were given in the classical books [12; 13; 37; 38].