## 3 The Procedure of the Numerical Solution
### The Applied 18 Numerical Algorithms
Then, a full time step is taken for the even cells (green box), and then a halved third timestep (blue rectangle) closes the repeating unit of the calculation, which is surrounded by a dashed red line. The golden rule is that the most recent values of the neighbors \(u_{i\pm 1}\) always have to be used when an \(A_{i}\) is calculated (for example, in the theta-formula) to obtain the new value of \(u_{i}\). This assures stability and, at the same time, quite quick convergence. At the next points, we give the concrete formulas with which the structures can be filled. In the case of the theta-formula, we just give the value of the parameter \(\theta\). 10. Original odd-even hopscotch method (OOEH, [43]): Stage 1: \(\theta=1\), Stage 2: \(\theta=0\). 11. Reversed odd-even hopscotch algorithm (RH, [34]), uses the same structure as the OOEH. Stage 1: \(\theta=0\), Stage 2: \(\theta=1\). 12. OEH-CNe method: OOEH structure, CNe formula in each stage with the golden rule mentioned above. 13. Shifted-hopscotch (SH, [33]): Stage 1 (yellow box in Figure 4): \(\theta=0\), Stages 2-4 (green boxes): \(\theta=\frac{1}{2}\), Stage 5 (blue box): \(\theta=1\). 14. Asymmetric-hopscotch (ASH, [44]): Stage 1 (yellow box), \(\theta=0\), Stage 2: \(\theta=\frac{1}{2}\), Stage 3: \(\theta=1\). \[u_{i}^{n+1}=\frac{(1-r_{i}\;\theta)u_{i}^{n}+A_{i}}{1+r_{i}\;(1-\theta)}\;,\; \theta\in[0,\;\;1] \tag{10}\]6 Hopscotch-type space-time structures. The time elapses from the top (\(t=0\)) to the bottom. 15. Leapfrog-hopscotch (LH, [45]): Stage 0 (yellow box): \(\theta=0\). Intermediate and last stages (green and blue boxes): \(\theta=\frac{1}{2}\). 16. Leapfrog-hopscotch-CNe (LH-CNe, [45]): CNe formula at all stages with the appropriate time-step size. 17. The Dufort-Frankel (DF) explicit two-step method [46] (p. 313) is an old but non-traditional algorithm which was constructed for the diffusion equation for which it is unconditionally stable. In our case, it has the formula: \[u_{i}^{n+1}=\frac{(1-r_{i}\;\theta)u_{i}^{n}+A_{i}}{1+r_{i}\;(1-\theta)}\;,\; \theta\in[0,\;\;1] \tag{10}\]7  
The r. h. s. of the formula contain values of \(u\) at two time levels, so we need two initial conditions, the second one calculated by another algorithm. We employ the UPFD Scheme (11) for this purpose. 18. For comparison purposes, we use the so-called classical version of the fourth-order Runge-Kutta (RK4) method [47] (p. 737). If we apply it to our spatially discretized system, we have  
\[u_{i}^{n+1}=\frac{(1-r_{i}\;\theta)u_{i}^{n}+A_{i}}{1+r_{i}\;(1-\theta)}\;,\; \theta\in[0,\;\;1] \tag{10}\]8  
\[u_{i}^{n+1}=\frac{(1-r_{i}\;\theta)u_{i}^{n}+A_{i}}{1+r_{i}\;(1-\theta)}\;,\; \theta\in[0,\;\;1] \tag{10}\]9  
\[u_{i}^{n+1}=\frac{u_{i}^{n}+A_{i}^{n}}{1+r_{i}^{n}} \tag{11}\]0  
and finally  
\[u_{i}^{n+1}=\frac{u_{i}^{n}+A_{i}^{n}}{1+r_{i}^{n}} \tag{11}\]1  
With the exception of the UPFD, OOEH, DF, and RK4, all the methods were created by our research group. The verifications and almost always the analytical proofs are presented in the original papers, but several truncation errors are given in [48]. The UPFD and the CNe methods have been shown to have first-order convergence in the time-step size, CCL and CLL are third order, RK4 is obviously fourth order, and all other schemes are second order. All methods (except RK4, of course) are unconditionally stable for the linear heat or diffusion equation. This means that the CFL restrictions do not apply in their case. We underline once again that in the big camp of explicit methods, unconditional stability is not the rule but the exception. Note that the CNe, CpC, LNe, LNe3, LNe4, OOEH-CNe, and LH-CNe schemes are not simply stable, but their error is strictly limited by the maximum and minimum principles [49] (p. 87). This is a direct consequence of the fact that any new \(u_{i}^{n+1}\) value is the convex combination of the already known \(u_{i}^{n},\;u_{i-1}^{n},\;u_{i+1}^{n}\), etc., which excludes the increase in any unphysical oscillations even for extremely large time-step sizes.