## 6 Numerical Methods and Algorithms
### Models at the Macro-Scale
#### 6.1.2 Solution of the \"Discrete in Space\"  
Model  
The **Time Integration algorithm** The proposed space discretization methods allow the construction of similar discrete problems since they all lead to a generally non-linear system of algebraic equations in the following form:  
\[\mathbf{M}\dot{\mathbf{u}}+\mathbf{K}(\mathbf{u})\mathbf{u}=\mathbf{f} \tag{111}\]  
which represents an ordinary differential equation (ODE) with respect to the time variable. Depending on the type of the approximation of the time derivative present in the left term of Equation (111) and the discretization of the temporal domain \(\Theta\), different methods are available to solve the problem described by Equation (111), which is discrete in space and continuous in time. Assuming that \(\Theta\) is the time interval \(\Theta=[0,\ T]\) and that it is discretized by a set of \(N_{T}+1\) equally spaced points \(\left\{t_{n}\right\}_{n=0...N_{T}}\), where \(t_{n}=\Delta t*n\) and \(\Delta t=\frac{T}{N_{T}}\), then there exist different methods--called \"_Time Integration Schemes (TIS)_\"--to compute \begin{table}
\begin{tabular}{p{142.3 pt} p{142.3 pt} p{142.3 pt} p{142.3 pt}} \hline \hline
**Discretization Methods** & _“Pros_” & _“Cons_” & **Refs** \\ \hline
**Finite Element Method (FEM)** & & & \\ \hline
**Finite Difference Method (FDM)** & & & \\ \hline
**Finite Volume Method (FVM)** & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Comparison, from the numerical point of view, between space discretization methods in terms of their _”Pros and Cons_”. The table contains, by way of example, references to the literature already introduced in Section 3 and regarding the use of discretization methods (where such information is available). the approximation \(u_{n+1}\) of \(u\) at the time \(t_{n+1}\) as a function of just the approximation of \(u\) and \(u\) at the time \(t_{n}\) (explicit methods) or as a function of \(u\) and \(u\) at the times \(t_{n}\) and \(t_{n+1}\) (implicit methods). \[u_{n+1} =f(u_{n},\dot{u}_{n},) \text{explicit schema}\] \[u_{n+1} =f(u_{n},\dot{u}_{n},u_{n+1}) \text{implicit schema}\]  
See [142] for a classification of such schemes. Among the rich set of available TIS, a one-parameter family of methods, called the \(\alpha\)-family, is commonly used [143]. In this family of methods, a weighted average of time derivative \(u\) of a dependent variable \(u\) is approximated at two consecutive time steps by a linear interpolation of the values of the same variable \(u\) at the same two steps:  
\[(1-\alpha)\dot{u}_{n}+\alpha\dot{u}_{n+1}=\frac{u_{n+1}-u_{n}}{\Delta t}. \tag{112}\]  
By substituting Equation (112) into Equation (111), the following relation is obtained  
\[\bar{\mathbf{K}}_{n+1}(\mathbf{u}_{n+1})\mathbf{u}_{n+1}=\bar{f}_{n,n+1}, \tag{113}\]  
where  
\[\bar{\mathbf{K}}_{n+1}(\mathbf{u}_{n+1}) = \mathbf{M}+a_{1}\mathbf{K}_{n+1} \tag{114}\] \[\bar{f}_{n,n+1} = \Delta t[\alpha f_{n+1}+(1-\alpha)f_{n}]+\bar{\mathbf{K}}_{n}\mathbf{u}_{n}\] (115) \[\bar{\mathbf{K}}_{n} = \mathbf{M}-a_{2}\mathbf{K}_{n} \tag{116}\]  
and where  
\[K_{i} = \mathbf{K}(\mathbf{u}_{i}),\;\forall i=0,\ldots,N_{T}, \tag{117}\] \[f_{i} = \mathbf{f}(t_{i}),\;\forall i=0,\ldots,N_{T},\] (118) \[u_{n+1} =f(u_{n},\dot{u}_{n},) \text{explicit schema}\]0 (119) \[u_{n+1} =f(u_{n},\dot{u}_{n},) \text{explicit schema}\]1  
Since at time \(t=0\) (i.e., \(n=0\)), the right-hand side is computed using the initial values defined in the time boundary conditions, and since the vector \(\mathbf{f}\) is always known, for both times \(t_{n}\) and \(t_{n+1}\), then the approximation \(u_{n+1}\) of \(u\) at time \(t_{n+1}\) (when \(n=0,\ldots,N_{T}-1\)) can be computed by Algorithm 1. For different values of the parameter \(\alpha\), several well-known time approximation schemes are obtained:  
\(\alpha=0\)**, the forward difference Euler explicit scheme**.