## 3 Analytical Results  
The most widespread definition of the truncation error is the difference between the discretized and the exact _equation_([PERSON], 1994) (p. 31). For example, in the case of Equation (1) and the forward time central space (FTCS) method,  
\[\tau=D_{t}^{+}u-D_{x}^{2}u, \tag{14}\]  
where  
\[D_{t}^{+}u=\frac{u\left(x_{i}^{n}\,\ t^{n}+\Delta t\right)-u\left(x_{i}^{n}\,\ t^{n}\right)}{ \Delta t}\ \text{and}\ D_{x}^{2}u=\frac{u\left(x_{i}^{n}\ -\Delta x,\ t^{n}\right)-2u\left(x_{i}^{n}\,\ t^{n}\right)+u\left(x_{i}^{n}\ +\Delta x,\ t^{n}\right)}{ \Delta x^{2}}\]  
are the first-order forward and second-order central difference operators for the time and the space variable, respectively. The Taylor-series expansions of \(u\) are inserted into Equation (14). The function \(u\) is supposed to be the exact solution, thus the substitutions \(u_{t}=\alpha u_{\left(2x\right)}\), \(u_{\left(2t\right)}=\left(\alpha u_{\left(2x\right)}\right)_{t}=\alpha^{2}u_{ \left(4x\right)}\), \(u_{\left(3t\right)}=\alpha^{3}u_{\left(6x\right)}\) and so on can be used. The discretization error of \(D_{x}^{2}\) is: \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Abbrev.** & **Name of the Method** & **Recent** & **Known LTE** & **O** & **Conv. Comb.** \\ \hline
1. & UPFD & Uncond. positive finite difference & - & + & 1 & + \\
2. & CNe & Constant neighbor & + & + & 1 & + \\
3. & CpC & Two-stage iterated CNe & + & - & 2 & + \\
4. & CpCC & Three-stage iterated CNe & new & - & 1 & + \\
5. & LNe & Two-stage Linear neighbor & + & + & 2 & + \\
6. & LNe3 & Three-stage iterated CNe & + & - & 2 & + \\
7. & LNe4 & Four-stage iterated CNe & + & - & 2 & + \\
8. & CLL & Const.-Lin.-Lin. neighbor fract. step & + & + & 3 & - \\
9. & PI & Pseudo-implicit & + & - & 2 & - \\
10 & ADE & Alternating direction explicit & - & + & 2 & - \\
11 & DF & Dufort-Frankel & - & + & 2 & - \\
12 & OOEH & Original odd–even hopscotch & - &? & 2 & - \\
13 & RH & Reversed hopscotch & + & - & 2 & - \\
14 & OEH-CNe & OEH structure with CNe formulas & + & - & 2 & + \\
15 & SH & Shifted hopscotch & + & - & 2 & - \\
16 & SH-CNe & SH structure with CNe formulas & + & - & 2 & + \\
17 & ASH & Asymmetric hopscotch & + & - & 2 & - \\
18 & ASH-CNe & ASH structure with CNe formulas & + & - & 2 & + \\
19 & LH & Leapfrog-hopscotch & + & - & 2 & - \\
20 & LH-CNe & LH structure with CNe formulas & + & - & 2 & + \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of the methods and their abbreviations. \[\tau_{0}=\varepsilon_{0}=-\frac{\alpha}{12}u_{(4x)}\Delta x^{2}-\frac{\alpha}{360}u_ {(6x)}\Delta x^{4}-\frac{\alpha}{20,160}u_{(8x)}\Delta x^{6}+\text{h. o. t.}, \tag{15}\]  
The discretization error of the \(D_{t}^{+}\) operator depends only on \(\Delta t\), thus the space- and time-dependent terms in the truncation error can be clearly separated. This is not true in our cases, where there are terms containing the ratio of \(\Delta t\) and \(\Delta x\). Moreover, for multistage methods, an expression with a similar form as (15) cannot be obtained, or the method is not clearly a FDM at all, since the differentiation with respect to time is not approximated by a finite difference operator. Nevertheless, all of the examined methods are explicit, and thus they can almost always be expressed in the fully explicit form \(u_{i}^{n+1}=f\big{(}u_{i}^{n},\ u_{i\pm 1}^{n},\ldots\big{)}\), where there are no quantities at the new \(n\)+1 time level present in the right-hand side, but the \(f\) function has the parameter \(r\). Note that in the numerical codes, these forms are used. The Taylor series expansion can be performed at this level as well [30]. In this way, one obtains the difference between the numerical and the exact _solution_. Let us denote the truncation error obtained by choosing this way by \(\varepsilon\). To present a simple example, in the case of the UFFD method, \(\tau\) is obtained when the expansions are applied to Equation (2), while \(\varepsilon\) is obtained when it is applied to Equation (3).