## 5 Examples
### Allen-Cahn Equation
We can now start applying the proposed methodology, as described above in Section 3 and following Figure 5. That is, we follow the path of the flowchart from the top to the bottom. We first classify the hardware scale according to P1 in the figure. The given hardware architecture, that is, an 8-core CPU system, falls well below the establishedrecommendation for the threshold of partitioned problems, which is at least 50 workers. Therefore, there is no need from a hardware side for massive parallelism. Next, we investigate the problem scale with process P2. The problem is governed by one scalar equation and there are no submodels involved, as defined in Section 4.6. Concerning the length scales, the presented system exhibits one extra physical length scale and that is the width \(\xi\) of the diffuse interface. This extra physical length scale originates from the nonlinearity of the Allen-Cahn equation and complements the other length scales, such as the dimensions of the domain, as well as the grid spacing, both being more natural in the numerical solution of PDEs. This poses the issue of numerical resolution of the systems length scales, that is, both the domain dimensions, as well as the width \(\xi\) of the diffuse interface, need to be properly represented on the discrete numerical grid [61; 62]. However, the fact that the problem is quasi-one-dimensional restricts the computational demands of the scenario. We thus arrive at the first decision point D1, where we can negate the necessity for massive parallelism. The next process step, P3, involves classifying the problem at hand, following the definition given in Section 4.2. As dependent variables, we encounter the time \(t\) as well as the spatial components \(x\) and \(y\). The coefficient matrix \(a_{ij}\), summing up all leading coefficients of second derivatives, then becomes for the 2D case  
\[a_{ij}^{AC}= \begin{bmatrix}0&0&0\\ 0&-1&0\\ 0&0&-1\end{bmatrix} . \tag{21}\]  
In this case, enumerating the eigenvalues \(\lambda_{i}\) is trivial, since \(a_{ij}^{AC}\) is a diagonal matrix, and we have \(\lambda_{0}=0\), \(\lambda_{1}=-1\), \(\lambda_{2}=-1\). We thus find that one eigenvalue is zero since the temporal derivative is only of first order and all other eigenvalues are of the same sign. Therefore, Equation (18) is a second-order PDE of parabolic type and we can proceed in D2 with the left branch. Moving on in the decision process, we would next classify the problem domain in D3 given input I3. As we use an equispaced grid in 1D, the discretization is Cartesian; thus, solving the problem using finite difference would be the best choice. As there are no other fields to classify according to decision point D5, we conclude the decision process. Within the unified methodological framework, we implement both FD and CG schemes and the scenario is comparatively solved using both schemes. This allows us to compare the schemes concerning numerical resolution capabilities and to investigate differences in the mutual arithmetic complexity and their impact on efficiency. Evaluating the CG method requires the reformulation of Equation (18) in its weak form, though. The finite dimensional weak statement is, then, as follows: Find \(\phi_{h}\in V_{h}\), such that  
\[\int_{\Omega}\frac{1}{M}v_{h}\partial_{t}\phi_{h}\,dx+\int_{\Omega}\
abla v_{ h}\
abla\phi_{h}\,dx-\int_{\Omega}v_{h}f(\phi_{h})\,dx=0\quad\forall v_{h}\in V _{h}, \tag{22}\]  
where we have already assumed the solution and test function to lie in the finite-dimensional subspace \(V_{h}\). Equation (18) requires the discretization of the Laplacian as its only differential operator. The temporal derivative will be treated using the method of lines approach, that is, we solve a large system of spatially discretized ordinary differential equations.