## 3 The Procedure of the Numerical Solution
### The Applied 18 Numerical Algorithms
If one repeats the corrector step again based on the LNe3 values to further improve the accuracy, a four-stage formula arises, which is abbreviated as LNe4. 7. The CLL algorithm [41] is a modification of the LNe3 algorithm in order to achieve third-order temporal convergence. For this purpose, it uses fractional time steps during the first and second stages. Generally, the length at the first stage is \(h_{1}=ph\), \(\frac{2}{3}\leq p<2\), but at the second stage it is always \(h_{2}=2h/3\). In the first stage, the CNe formula is employed to calculate new predictor values:  
\[u_{i}^{\text{C}}=u_{i}^{n}e^{-\ p_{i}^{n}}+\frac{A_{i}^{n}}{r_{i}^{n}}\left(1 -e^{-\ p_{i}^{n}}\right) \tag{16}\]  
In the second stage, we use formulas similar to (15) but with an \(h_{2}=2h/3\) time-step size to obtain the first corrector values. The new \(A_{i}^{\text{new}}\) values are calculated as in Equation (14), i.e., \(A_{i}^{\text{C}}=\frac{h}{c_{i}}\bigg{(}\frac{u_{i-1}^{\text{C}}}{R_{i,\ i-1}^{\text{C}}}+\frac{u_{i+1}^{\text{C}}}{R_{i,\ i+1}^{\text{C}}}\bigg{)}\). Using these the corrector step is as follows:  
\[u_{i}^{n+1}=\frac{(1-r_{i}\;\theta)u_{i}^{n}+A_{i}}{1+r_{i}\;(1-\theta)}\;,\; \theta\in[0,\;\;1] \tag{10}\]0  
In the third stage, a full time step is taken with the LNe formula:  
\[u_{i}^{n+1}=\frac{(1-r_{i}\;\theta)u_{i}^{n}+A_{i}}{1+r_{i}\;(1-\theta)}\;,\; \theta\in[0,\;\;1] \tag{10}\]1  
where \(A_{i}^{\text{CL}}=\frac{h}{C_{i}}\bigg{(}\frac{u_{i-1}^{\text{CL}}}{R_{i,\ i-1}^{\text{C}}}+\frac{u_{i+1}^{\text{CL}}}{R_{i,\ i+1}^{\text{C}}}\bigg{)}\). In Section 4, we take \(p=\frac{2}{3}\), but in Section 5 we try \(p=1\) as well. 8. The CCL algorithm [42] is very similar to the CLL, but at the second stage the CNe formula is used. So, the first stage is the same as in (16), where, for stability reasons, \(\frac{1}{3}\leq p\leq 0.949\) and \(p\
eq\frac{2}{3}\) should hold. We set \(p=\frac{1}{3}\) and obtain the first- and second-stage formulas:  
\[u_{i}^{n+1}=\frac{(1-r_{i}\;\theta)u_{i}^{n}+A_{i}}{1+r_{i}\;(1-\theta)}\;,\; \theta\in[0,\;\;1] \tag{10}\]2  
Then, after the calculation of \(A_{i}^{\text{CC}}=\frac{h}{C_{i}}\bigg{(}\frac{u_{i-1}^{\text{CC}}}{R_{i,\ i-1}^{\text{CL}}}+\frac{u_{i+1}^{\text{CC}}}{R_{i,\ i+1}^{\text{C}}}\bigg{)}\), a full time step is taken with the LNe formula at the third stage:  
\[u_{i}^{n+1}=\frac{(1-r_{i}\;\theta)u_{i}^{n}+A_{i}}{1+r_{i}\;(1-\theta)}\;,\; \theta\in[0,\;\;1] \tag{10}\]39. The two-stage pseudo-implicit (PI) method is published in [38]. It is considered here with parameters \(p=\frac{1}{2}\) and \(\theta_{1}=0\) at the first stage and \(\theta_{2}=\frac{1}{2}\) at the second stage. The following first- and second-stage formulas must be applied for each cell:  
\[u_{i}^{n+1}=\frac{(1-r_{i}\;\theta)u_{i}^{n}+A_{i}}{1+r_{i}\;(1-\theta)}\;,\; \theta\in[0,\;\;1] \tag{10}\]4  
where \(A_{i}^{\text{new}}\) is calculated exactly as in Equation (13). One can see that there is a stage with a half time step for the predictor values and then the second stage with a full time step for the corrector values. Let us now focus on the odd-even hopscotch schemes. A bipartite spatial grid, in which all the nearest neighbors of the odd cells are even and vice versa, is required to apply any of them. There are four different stencils, more precisely spatial and temporal structures, of the used hopscotch-type methods, which are displayed in \[u_{i}^{n+1}=\frac{(1-r_{i}\;\theta)u_{i}^{n}+A_{i}}{1+r_{i}\;(1-\theta)}\;,\; \theta\in[0,\;\;1] \tag{10}\]5 Only one odd and one even cell is displayed in the case of each method in the figure. The stages are symbolized by colored rectangles. Those boxes which belong to the repeating units are surrounded by dashed red lines. For example, the asymmetric hopscotch structure (ASH) consists of two half and one full time steps. The calculation starts with a half-sized time step (yellow rectangle with the number '1' inside) being taken for the odd cells using the initial values.