## 3 Results and Discussion
### Accuracy of the Numerical Solution  
Finally, we discuss the accuracy of the proposed method. In particular, we compare the cooling curve for different configurations discussed above. We choose one node in the cast and extract the temporal evolution of two quantities: temperature and the solid phase fraction. We plot both quantities of interest in Figures 8 and 9. We compare four various combinations of time integration schemes for the case \(m=1\) (the standard and partitioning methods). We notice only small differences in temperature and solid phase profile between tested cases. All of them appear under the eutectic temperature where the casting is completely solidified, and a cooling process occurs. We observe the largest differences (the highest standard deviation) for the combination of \(E-E^{m}\). However, these differences are small (for \(m=15\) even smaller) and do not exceed a few percents, which is acceptable. \begin{table}
\begin{tabular}{c c c c} \hline  & \(\mathbf{m=1}\) & \(\mathbf{m=15}\) \\ \hline  & \(t,s\) & \(p_{m}\) & \(p_{m}\) \\ \hline \(I-I^{m}\) & 3841.51 & 1 & 2.5 \\ \(E-E^{m}\) & 2340.73 & 1 & 1.9 \\ \(E-I^{m}\) & 3268.53 & 1 & 3.1 \\ \(I-E^{m}\) & 3557.40 & 1 & 2.3 \\ \hline \end{tabular}
\end{table} 
Table 7: An efficiency \(p_{m}\) obtained with the mixed time partitioning methods. We further test the accuracy of the sub-cycles and mixed time partitioning methods (\(m=15\)). We plot corresponding profiles of temperature and solid fractions in Figures 10 and 11. The shape of both presented curves indicates that between the liquidus and solidus temperatures solidification process occurs expectedly. Moreover, the distribution of temperature and solid phase fraction obtained for multiplier \(m=15\) is consistent with the above observations. The differences appearing between applied schemes are smaller than mentioned above one (for \(m=1\)). The benefits of using mixed time partitioning methods outweigh the slight loss of the results' accuracy (Figure 12). Figure 8: Cooling curves obtained for selected point in casting domain. Temperature distribution for multiplication factor \(m=1\) calculated by mixed time partitioning methods. Figure 9: Solid phase growth obtained for selected point in casting domain. Solid phase fraction for multiplication factor \(m=1\) calculated by mixed time partitioning methods. Figure 11: Solid phase growth obtained for selected point in casting domain. Solid phase fraction for multiplication factor \(m=15\) calculated by mixed time partitioning methods. Figure 10: Cooling curves obtained for selected point in casting domain. Temperature distribution for multiplication factor \(m=15\) calculated by mixed time partitioning methods. Figure 14: Solid phase fraction in whole casting domain. Four combination of the time integration schemes (**A**) \(I-I^{m}\), (**B**) \(E-E^{m}\), (**C**) \(E-I^{m}\), (**D**) \(I-E^{m}\) for multiplication factor \(m=1\). Figure 15: Temperature field in whole domain (a shade of red color—the highest temperature, connected with the cast and a blue color—the lowest temperature). Four combination of the time integration schemes (**A**) \(I-I^{m}\), (**B**) \(E-E^{m}\), (**C**) \(E-I^{m}\), (**D**) \(I-E^{m}\) for multiplication factor \(m=15\). In the final analysis, we may see that the use of implicit time integration method increases the computational time but allows to use larger time step. However, one should be careful as too large size of time step may lead to the oversight of some important physical phenomena (e.g., solid phase fraction) or lead to significant errors. We purposefully chose the multiplication factor \(m\) larger than the acceptable value (determine in accordance with stability criterion described in Section 2.1.2) and indeed got erroneous results--the incorrect distribution of the temperature and solid phase fraction (see Figure 16). In conclusion, we showcased that using mixed time partitioning methods indeed offers significant gains.