## 2 The Analyzed Numerical Methods
* The CLL method [33] is very similar to the LNe3 method, but it applies fractional time steps at the first and second stages with step sizes \(\Delta t_{1}=p\Delta t\), \(\Delta t_{2}=p_{2}\Delta t\). Due to these, it achieves third-order convergence in the time step size, but only if \(p_{2}=\
icefrac{{2}}{{3}}\). If the length of the second stage is not fixed to this value, then low-order expressions with coefficient \(3p_{2}-2\) appear in the truncation errors, which make the method second-order only. In this paper, we fixed \(p_{2}=\
icefrac{{2}}{{3}}\) and keep \(p\) as a free parameter. Therefore, the first stage applies the following formula: \[u_{i}^{C}=u_{i}^{n}\cdot e^{-2 pr}+\frac{u_{i-1}^{n}+u_{i+1}^{n}}{2}\left(1-e^ {-2 pr}\right)\] In the second stage, a formula similar to (6) is used, but with a reduced time step size: \[u_{i}^{\text{CL}}=u_{i}^{n}e^{-4r/3}+\frac{u_{i-1}^{n}+u_{i+1}^{n}}{2}\Big{(}1 -e^{-4r/3}\Big{)}+\frac{s_{i}^{1}}{2p}\Bigg{(}\frac{2}{3}-\frac{1-e^{-4r/3}}{ 2r}\Bigg{)},\] (7) where \(s_{i}^{1}=u_{i-1}^{C}+u_{i+1}^{C}-u_{i-1}^{n}-u_{i+1}^{n}\). In the third stage, a full time step is taken: \[u_{i}^{n+1}=\frac{u_{i}^{n}+r\big{(}u_{i-1}^{n}+u_{i+1}^{n}\big{)}}{1+2r},\]0 (8) where \(s_{i}^{2}=u_{i-1}^{\text{CL}}+u_{i+1}^{\text{CL}}-u_{i-1}^{n}-u_{i+1}^{n}\). * The pseudo-implicit (PI) two-stage method was developed in our previous publication [14] (Algorithm 5 there) for the conduction-convection-radiation equation. Here, we apply it only to the pure diffusion Equation (1) with parameter \(\lambda=1\), which means that a half time step is taken to obtain the predictor values and then a full time step for the corrector values. The formulas are the following: Stage 1: \(u_{i}^{\text{pred}}=\frac{u_{i}^{n}+\frac{1}{2}\big{(}u_{i-1}^{n}+u_{i+1}^{n} \big{)}}{1+r}\) Stage 2: \(u_{i}^{n+1}=\frac{(1-r)u_{i}^{n}+r\big{(}u_{i-1}^{\text{pred}}+u_{i+1}^{\text {pred}}\big{)}}{1+r}\), Note that the trick of the implicit and explicit treatment of the actual node and its neighbors is the same as in the UPFD method, so this PI method can actually be considered as a generalization of the UPFD scheme. * The alternating direction explicit (ADE) scheme is a known [26; 43] but non-conventional method for which the condition of consistency is also known. We include it here for comparison purposes. In a one-dimensional equidistant mesh, one splits the calculation, i.e., first sweeps the mesh from the left to right, and then vice versa. In the case of Dirichlet boundary conditions at nodes 0 and \(N\), one sets: \[u_{i}^{n+1}=\frac{u_{i}^{n}+r\big{(}u_{i-1}^{n}+u_{i+1}^{n}\big{)}}{1+2r},\]1 Then, the following equations are solved from left to right and from right to left, respectively: \[u_{i}^{n+1}=\frac{u_{i}^{n}+r\big{(}u_{i-1}^{n}+u_{i+1}^{n}\big{)}}{1+2r},\]2 (9) The explicit expressions are the following: \[u_{i}^{n+1}=\frac{u_{i}^{n}+r\big{(}u_{i-1}^{n}+u_{i+1}^{n}\big{)}}{1+2r},\]3 (10)The final values are the simple averages of the two half-sided terms: \(u_{i}^{n+1}=(p_{i}^{n+1}+q_{i}^{n+1})/2\). If one expresses the final value of \(u\), one obtains:  
\[u_{i}^{n+1}=\frac{u_{i}^{n}+r\big{(}u_{i-1}^{n}+u_{i+1}^{n}\big{)}}{1+2r},\]4  
which is actually the same formula as in the Crank-Nicolson method, though the solution of this system of equations is completely different in the two methods. We note that for non-uniform meshes, the ADE method loses its fully explicit character and matrix calculations will be necessary. 11. The Dufort-Frankel (DF) method [44] (p. 313) is the textbook example of explicit and unconditionally stable methods. It is a one-stage but two-step algorithm with the formula: \[u_{i}^{n+1}=\frac{u_{i}^{n}+r\big{(}u_{i-1}^{n}+u_{i+1}^{n}\big{)}}{1+2r},\]5  
This method is not a self-starter, and thus \(u_{i}^{1}\) has to be calculated from \(u_{i}^{0}\) using another method. We use two half-sized UFFD time step for this purpose. For the application of odd-even hopscotch methods, the space must be discretized by a special, so-called bipartite grid.