## 2 The Examined Numerical Methods
### Methods Used for Comparison Purposes  
We present three explicit methods, which are known to be unconditionally stable for the heat conduction case. However, as far as we know, they have not been applied to the case where convection and radiation are also present. 1. The Dufort-Frankel (DF) method [34] (p. 313) is the textbook example of explicit and unconditionally stable methods. It is a two-step but one-stage algorithm with the following formula, where the convection and the radiation terms are treated in a mixed way:  
\[u_{i}^{n+1}=\frac{(1-2r)u_{i}^{n-1}+2r\big{(}u_{i-1}^{n}+u_{i+1}^{n}\big{)}+2 hq _{i}-hKu_{i}^{n}-h\sigma\big{(}u_{i}^{n}\big{)}^{4}}{1+2r+hK+h\sigma\big{(}u_{i }^{n}\big{)}^{3}} \tag{25}\]  
Since this algorithm is not a self-starter, \(u_{i}^{1}\) must be calculated from \(u_{i}^{0}\) by another method. We employ the UPFD formula [31] for this purpose:  
\[u_{i}^{n+1}=\frac{u_{i}^{n}+r\big{(}u_{i-1}^{n}+u_{i+1}^{n}\big{)}+hq_{i}}{1+2 r+hK+h\sigma\big{(}u_{i}^{n}\big{)}^{3}}\]  
2. The alternating direction explicit (ADE) scheme is a known, but non-conventional method [23; 35]. In a one-dimensional equidistant mesh, one splits the calculation into two directions, first sweeping the mesh from the left to right (using auxiliary variable \(a\)) and then vice versa (with variable \(b\)). In the case of Dirichlet boundary conditions at nodes 0 and \(N\), one sets  
\[a_{i}^{n}=u_{i}^{n}\,\ i=1,\ldots N\,\ a_{0}^{n+1}=u_{0}^{n+1}\ \text{and}\ b_{i}^{n}=u_{i}^{n}\,\ i=N,N-1,...1\,\ b_{N}^{n+1}=u_{N}^{n+1}\]  
Then, in case of pure conduction, the following equations are solved from left to right and from right to left, respectively:  
\[\frac{a_{i}^{n+1}-a_{i}^{n}}{h}=\frac{\alpha}{\Delta x^{2}}\Big{(}a_{i-1}^{n+1 }-a_{i}^{n+1}-a_{i}^{n}+a_{i+1}^{n}\Big{)}\]  
and  
\[\frac{b_{i}^{n+1}-b_{i}^{n}}{h}=\frac{\alpha}{\Delta x^{2}}\Big{(}b_{i-1}^{n} -b_{i}^{n+1}-b_{i}^{n}+b_{i+1}^{n+1}\Big{)}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline  & & & **Equation Number** \\  & & & **or Point** \\ \hline LH FullExp & Fully explicit & convection, Section 2.2 & (8) \\ \cline{3-4}  & & radiation, Section 2.3 & (16) \\ \cline{3-4}  & & convection, Section 2.2 & (11) \\ \cline{3-4}  & QuasiEx & & radiation, Section 2.3 & (18) \\ \cline{3-4}  & & convection, Section 2.2 & (13) \\ \cline{3-4}  & & radiation, Section 2.3 & (20) \\ \cline{3-4}  & & convection, Section 2.2 & (14) \\ \cline{3-4}  & & radiation, Section 2.3 & (21) \\ \hline LH Inside & Inside with the & convection, Section 2.2 & (14) + (24) \\ \cline{3-4}  & non-negative trick & radiation, Section 2.3 & (21) + (24) \\ \hline LH Mixed & \begin{tabular}{c} Mixture of the \\ pseudo-implicit and inside \\ with the weight of the PI \\ \end{tabular} & convection, Section 2.2 & (15) \\ \cline{3-4} LH PI NL prod & \begin{tabular}{c} Pseudo-implicit, nonlocal \\ with product \\ \end{tabular} & radiation, Section 2.3, 1D & point 6 \\ \hline LH PI NL av & \
\begin{tabular}{c} Pseudo-implicit, nonlocal \\ with space-average \\ \end{tabular} & radiation, Section 2.3, 1D & point 7 \\ \cline{3-4}  & & radiation, Section 2.3, 2D & (39) \\ \hline LH PI NL time &
\begin{tabular}{c} Pseudo-implicit, nonlocal \\ with time-average \\ \end{tabular} & radiation, Section 2.3 & point 8 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The different treatments of the convection and the radiation terms and the defining equations. Since, on the right hand side of these formulas, both \(a_{i}\) and \(b_{i}\) are taken into account 50-50% in the old and new time level, it is plausible to use the mixed treatment of the convection and radiation term here, too.