Fig. 4

Convergence test for numerically solving Eq. (12). The numerical error \(\left| {{\Delta }T} \right|\) was evaluated at \(\left( {x,t} \right) = \left( {0 \;{\text{m}},6 \;{\text{s}}} \right)\) and plotted against element size \({\Delta }x\). The numerical solutions were obtained by the explicit Runge–Kutta method of order 5(4) (see main text for details). All parameters, except for \({\Delta }x\), are the same as those used the numerical solution shown in Fig. 3b and listed in Table 1