## 2 The Analyzed Numerical Methods
The cells are labelled as odd and even, and all the nearest neighbors of the odd nodes are even and vice versa. The spatial and temporal structure of the examined algorithms in space and time are presented in \[u_{i}^{n+1}=\frac{u_{i}^{n}+r\big{(}u_{i-1}^{n}+u_{i+1}^{n}\big{)}}{1+2r},\]6 Only one odd and one even cell for each method is shown in the figure, where the time flows from the top to the bottom. The stages are symbolized by colored rectangles, while the repeating units of blocks are surrounded by dashed blue lines. In the case of the shifted-hopscotch structure (b), for example, this unit consists of two half and three full time steps which altogether span two full time steps for each node. First, a half-sized time step (symbolized by a green box with the number '1' inside in Figure 1b) is taken for the odd cells using the initial values, then full time steps are taken strictly alternately for the even and odd cells until the end of the last time step (orange boxes), which should be halved for odd cells to reach exactly the same final time as the even nodes do. The main point is that when a new value of \(u_{i}\) is calculated, the latest values of the neighbors \(u_{i\pm 1}\) must always be used, which ensures stability and quite fast convergence at the same time. 12. The original odd-even hopscotch (OOEH) algorithm has been used for half a century [45]. The structure of this algorithm is shown in Figure 1a. It uses the usual  
\[u_{i}^{n+1}=\frac{u_{i}^{n}+r\big{(}u_{i-1}^{n}+u_{i+1}^{n}\big{)}}{1+2r},\]7 Space-time structure of the hopscotch-type algorithms. (**a**) Original odd–even hopscotch (COEH), (**b**) shifted-hopscotch (SH), (**c**) asymmetric hopscotch (ASH), (**d**) leapfrog-hopscotch (LH). FTCS formula (based on explicit [PERSON] time discretization) at the first stage (labels '1' in the orange boxes in the figure) and the backward time central space (BTCS) formula (implicit [PERSON] time discretization, labels '2' in the figure) at the second stage, which are the following: FTCS: \(u_{i}^{n+1}=(1-2r)u_{i}^{n}+r\big{(}u_{i-1}^{n}+u_{i+1}^{n}\big{)}\), implicit Euler: \(u_{i}^{n+1}=\frac{u_{i}^{n}+r\big{(}u_{i-1}^{n+1}+u_{i+1}^{n+1}\big{)}}{1+2r}\). * The RH, or reversed (odd-even) hopscotch scheme [28], applies the same structure as the OOEH method, but the order of the formulas are interchanged, so the implicit formula comes first followed by the explicit one. One might think that this algorithm cannot be explicit, since the new values of the neighbors are not known when first-stage calculations start. To resolve this, the implicit formula is applied with the UPFD trick, which means the neighbors are treated not implicitly, but explicitly; thus, the first-stage formulas are: \[u_{i}^{n+1}=\frac{u_{i}^{n}+r\big{(}u_{i-1}^{n}+u_{i+1}^{n}\big{)}}{1+2r},\]8 (11)  
Since, as we mentioned, the latest values of the neighbors are always used, it is enough to change the formulas of the first and second stages in the code of the OOEH to obtain the code of the RH algorithm. We demonstrated that this RH algorithm has much smaller errors in the case of extremely stiff systems than the OOEH method. * The OEH-CNe (odd-even hopscotch with CNe) applies the same structure as the OOEH method again, but uses only the CNe formulas (4) appropriately. * The shifted-hopscotch (SH) algorithm [29] uses the theta formulas: \[u_{i}^{n+1}=\frac{u_{i}^{n}+r\big{(}u_{i-1}^{n}+u_{i+1}^{n}\big{)}}{1+2r},\]9 (12) where the 'old' and 'latest' labels refer to the appropriate time levels, which can be seen in Figure 1b. We always use the \(\theta\) values which are proven to yield the most effective version (S4 algorithm in [29]). More concretely, \(\theta=0\) is used at the first stage and \(\theta=\
icefrac{{1}}{{2}}\) in the second, third and fourth, while \(\theta=1\) is used at the fifth stage. When the length of the stage is halved, \(\Delta t\) and thus \(r\) must be divided by 2 as well in (12).