## 3 Analytical Results
In the latter case, the \((1+2r)^{-1}\) coefficient must also be expanded and the result will be different from \(\tau\). One of the sources of the difference is that when Equation (3) is obtained from Equation (2), we multiplied it by \(\Delta t\), which is quite common; see p. 20 in [37]. Therefore, \(\varepsilon\) must be divided by \(\Delta t\) to make the two kinds of errors comparable. In fact, after this division, both \(\tau\) and \(\varepsilon\) give an estimation for the global error [30]. However, we will see that \(\tau\) and \(\varepsilon\) are usually still different and it is a nontrivial question which one reflects more the properties of the numerical scheme, i.e., which one is more useful for practical purposes. The term (15) is present in case of all the examined methods, both in \(\tau\) and in \(\varepsilon\). We note that the higher order terms (h. o. t.) will be usually omitted for the sake of brevity. The \(\tau\) and \(\varepsilon\) errors have been calculated in most cases by both the first and the second author completely independently. If one follows the so-called method of lines, where Equation (1) is discretized only spatially using the central difference formula, one obtains the ODE system:  
\[\frac{d\overrightarrow{u}}{dt}=M\overrightarrow{u}\,\ \overrightarrow{u}(t=0)= \overrightarrow{u}^{0}, \tag{16}\]  
where \(\overrightarrow{u}^{0}\) is the arbitrary initial vector. The elements of \(M\) can be given as:  
\[m_{ii}=-\frac{2\alpha}{\Delta x^{2}}\,\ \ \ \ m_{i,i+1}=m_{i,i-1}=\frac{\alpha}{ \Delta x^{2}}\ \ \ (1<i<N),\]  
while the first and last row are determined by the boundary condition. The exact solution of (16) at the end of a time step is the following:  
\[\overrightarrow{u}^{n+1}=\text{e}^{M\Delta t}\overrightarrow{u}^{n}=\bigg{(}1 +M\Delta t+M^{2}\frac{\Delta t^{2}}{2}+M^{3}\frac{\Delta t^{3}}{3!}+O\Big{(} \Delta t^{4}\Big{)}\bigg{)}\overrightarrow{u}^{n}.\]  
Here, as well as in the case of the \(e^{-2r}\)-type terms in the CNe, LNe, etc., methods, the usual series expansion \(e^{\pm x}=1\pm x+x^{2}/2!\ \pm x^{3}/3!\ +O\big{(}x^{4}\big{)}\) is used up to the appropriate order. After some simple algebraic calculations, we obtained the expressions of a general element of the exact solution after one and two time steps:  
\[u_{i}^{n+1}=\frac{r^{2}}{2}u_{i\pm 2}^{n}+\Big{(}r-2r^{2}\Big{)}u_{i\pm 1}^{n}+ \Big{(}1-2r+3r^{2}\Big{)}u_{i}^{n}+O\Big{(}r^{3}\Big{)}, \tag{17}\]  
and  
\[u_{i}^{n+2}=2r^{2}u_{i\pm 2}^{n}+\Big{(}2r-8r^{2}\Big{)}u_{i\pm 1}^{n}+\Big{(}1 -4r+12r^{2}\Big{)}u_{i}^{n}+O\Big{(}r^{3}\Big{)}\, \tag{18}\]respectively. The simple notation \(u^{n}_{i\pm j}=u^{n}_{i+j}+u^{n}_{i-j}\), \(i,j\in\mathbb{N}^{+}\) has been introduced for brevity. Formulas (17) and (18) are valid if we are far enough from the boundaries. To ensure this, we always examine the middle element of a large enough matrix. A similar expression can be obtained for each numerical method. More precisely, for the hopscotch methods, we have different formulas for the odd and the even nodes. If the smallest repeating unit spans one or two time steps, then Equation (17) or (18) must be used, respectively. The local error is the difference between the analytical and the numerical expressions, which is divided by \(\Delta t\) again to obtain the global error, which is denoted by \(\delta\). We say that a method is, e.g., second-order if the zeroth- and the first-order terms vanish in \(\delta\), and the second-order terms have the lowest order. This kind of investigation has been performed for some of our methods in the original publications, but only in order to prove the order of the method. The first non-vanishing terms are given for the first time here, which is also a novelty in the current paper. The calculated errors are presented in the Appendix A.