## 2 Numerical Model of Solidification  
Solidification is described by a quasi-linear heat conduction equation with additional heat source term along with the boundary and initial conditions:  
\[\
abla\cdot(\lambda\
abla T)+\rho_{s}L\frac{\partial f_{s}}{\partial t}=c\rho \frac{\partial T}{\partial t}, \tag{1}\]  
\[\Gamma_{1}:\ \ q=\alpha(T)(T_{|\Gamma_{1}}-T_{emp}(t)), \tag{2}\]  
\[\Gamma_{2}:\left\{ \begin{array}{l}-\lambda_{1}\frac{\partial T^{(1)}}{ \partial\mathbf{n}}=-\lambda_{2}\frac{\partial T^{(2)}}{\partial\mathbf{n}}= \frac{\beta}{\delta}(T^{(1)}-T^{(2)}),\\ T^{(1)}\
eq T^{(2)},\end{array} \right. \tag{3}\]  
where \(\lambda\) is the thermal conductivity coefficient, \(c\) is the specific heat, \(\rho\) is the density (subscript \(s\) refers to the solid phase), \(L\) is the latent heat of solidification, \(f_{s}\) is the solid phase fraction, \(q\) is the heat flux, \(t\) is the time and \(T\) is the temperature. Boundary condition given by Equation (2) is the Newton boundary condition, where the exchange of heat on \(\Gamma_{1}\) with the environment takes place  
Figure 1: Transition from partial differential equation to approximate solution by using mixed time partitioning methods. Figure 2: Origin of mixed time partitionig methods. with heat transfer coefficient \(\alpha\). \(T_{|\Gamma_{1}}\) is the temperature on the \(\Gamma_{1}\) boundary of the domain and \(T_{env}\) is the environment temperature. Boundary condition given by Equation (3) is the continuity boundary condition on the \(\Gamma_{2}\) boundary separating \(\Omega_{1}\) and \(\Omega_{2}\) domains with temperature \(T^{(1)}\) and \(T^{(2)}\), respectively. The thermal conductivity coefficient of the material in the separating layer is given by \(\beta\), and \(\delta\) is the thickness of this layer and \(\mathbf{n}\) is the normal vector to the boundary. In this work, we use the apparent heat capacity formulation (\(c^{*}\)) of solidification:  
\[\
abla\cdot(\lambda\
abla T)=c^{*}(T)\frac{\partial T}{\partial t}, \tag{4}\]  
\[c^{*} = c\rho_{s},\quad\text{for}\quad T<T_{s},\] \[c^{*} = c\rho_{f}-\rho_{s}L\frac{\partial f_{s}}{\partial T},\quad\text{ for}\quad T_{s}\leq T\leq T_{l}, \tag{5}\] \[c^{*} = c\rho_{l},\quad\text{for}\quad T>T_{l}.\]  
where subscript \(l\) refers to the liquid phase, \(T_{s}\) is the solidus temperature (the end of solidification) and \(T_{l}\) is the liquids temperature (the beginning of solidification). We used indirect model of solid phase growth for two-component alloy:  
\[f_{s}(T)=\frac{1}{1-\eta k\varepsilon}\left(1-\left(\frac{T_{M}-T}{T_{M}-T_{l} }\right)^{\frac{1-\eta k}{k-1}}\right), \tag{6}\]  
where \(\eta\) is the coefficient describing the grain's shape, \(k\) is the solute partition coefficient, \(T_{M}\) is the solidification temperature of the pure basic component of an alloy, and \(\varepsilon\) is the so-called Brody-Flemings coefficient. For more details, we refer to [14; 15]. All the above equations form the thermal description basis of solidification. Generally speaking, to solve this initial boundary value problem we first introduce spatial discretization and then time discretization (see Figure 1). Firstly, we use the finite element method ([16; 17; 18; 19; 20; 21] et al.) to obtain the ordinary differential equation, given as:  
\[\mathbf{M}(T)\dot{\mathbf{T}}+\mathbf{K}(T)\mathbf{T}=\mathbf{b}(T), \tag{7}\]  
where \(\mathbf{M}\) is the capacity matrix, \(\mathbf{K}\) is the conductivity matrix, \(\mathbf{T}\) is the temperature vector and \(\mathbf{b}\) is the right-hand side vector, whose values are calculated on the boundary conditions basis. Secondly, we apply time steppers to solve ordinary differential equation. This step is the major element of this work.