## 5 Examples
### Allen-Cahn Equation
We once again follow the process summarized in Figure 5. Regarding the system of PDEs (I1), Equation (27) is simply an algebraic constraint and thus can be calculated in a simple postprocessing step. Thus, (27) is not a governing equation in the sense of a PDE, and \(\alpha_{2}\) will consequently not be considered an independent variable, as detailed in Section 4.2. We are then left to solve a single scalar advection equation for \(\alpha_{1}\). Proceeding in the flow chart, we next classify the hardware scales within P1. Here, we find that the last hardware configuration listed in Table 4 necessitates the use of schemes that are tailored for high parallelism, as the given amount of 128 processes is above the specified regime where the use of parallelizable algorithms is worthwhile using. Therefore, this configuration should be run using a discontinuous Galerkin method. For both other configurations using 8 and 18 processes, this does not apply. Continuing with process P2, we find that the given problem only exhibits one length scale; thus, this criterion for parallelism can be omitted. Thus, we arrive at decision D1 and find that for the problem statement involving the largest of the three computing architectures, the use of the discontinuous Galerkin method is advised. For the remaining two configurations, we can proceed by classifying the PDE according to process P3. With the temporal derivative and gradient as the only differential operators, Equation (1) is a first-order PDE. The advection velocity vector \(\underline{u}\) has constant and real components. Thus, following Section 4.2, we find that the advection equation presented here is hyperbolic and proceed with the right branch of the flow chart after decision D2. Consequently, we need to evaluate the linearity of Equation (26) for process P4 as a next step. As the terms including the differential operators are linear and there is no right-hand side, it may be classified straightforwardly to be linear. Due to its linearity, the most efficient choice for the remaining configurations turns out to be the discontinuous Galerkin method as well. As stated previously, the original two-equation system only consists of one PDE; thus, we conclude the decision process here, as all fields governed by a PDE have been assigned (decision D5). It should be noted here that when the VoF method is used together with the finite volume method, one can expect the interface of two phases to diffuse considerably. Within the numerics community, this problem is well known and has led to the practice of introducing an additional interface compression term into the advection equation [87]. \[\partial_{t}\alpha+\underline{u}\cdot\
abla\alpha+\
abla\cdot[ \underline{u}_{c}\alpha(1-\alpha)]=0, \tag{30}\] \[\underline{u}_{c}=\min(c_{\alpha}|\underline{u}|,\max(| \underline{u}|))\frac{\
abla\alpha}{|\
abla\alpha|}, \tag{31}\]

where the newly introduced, artificial interface compression velocity \(\underline{u}_{c}\) is used. In this case, \(c_{\alpha}\) is a positive, scalar constant somewhere around unity. As can easily be observed by combining Equations (30) and (31), this makes the resulting PDE fully nonlinear, which can be regarded as a severe disadvantage in this case. This additional, artificial interface motion counteracts the diffusion that the finite volume method has shown to exhibit. As \begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**CPU Name** & **Number of** & **Core Clock** & **Memory Size** & **Memory Bandwidth** & **Memory Speed [GHz]** \\ \hline Apple M1 & 8 & 3.2 & 16 & 200 & 6.4 \\ Pro & 18 & 3.0 & 128 & 94 & 2.9 \\
2x AMD & 128 & 2.45 & 512 & 204 & 3.2 \\ EPYC 7763 & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Hardware configurations for the advection equation model problem. The three setups mimic popular computing environments in applied settings: a mobile computer, a stationary workstation grade tower, and a server tailored to numerical computing.