## 5 Examples
### Allen-Cahn Equation
This relationship is visualized for reference in Figure 10.

The benchmarking results for all three hardware configurations are given in Figure 11. We report the total run times as well as the solution time for one singular time step. This is since, as explained, the DG model has a considerably smaller admissible time step. Thus, comparing overall run times is not suitable to empirically validate the above claims regarding parallel efficiency. The overall computation time is, nonetheless, an important factor in terms of practicality since this is the main quantity one is interested in when performing a simulation. It becomes evident from the reported wall times that the FV model only has an advantage on the small desktop machine concerning solution time. As predicted, scaling up the problem size and number of workers will yield an increasing advantage for the higher-order DGM. The differences in computation time which were found to be empirically

Figure 11: Comparison of run times for the DGM and FVM advection benchmark case, run on the three configurations given in Table 4. _Left:_ Weak scaling of both models, measured as wall time per time step solve. The number of DoFs/core was fixed at 18,400. For each sample population, \(n\approx 3000\). Horizontal lines: median, boxes: IQR, whiskers: quartiles \(\pm 1.5\) IQR, ****: \(p<0.0001\). _Right:_ Total wall times for the solution over all time steps. Figure 10: Comparison of shared DoFs (colored dots) between cells for a fourth-order DG method with an FV method with an equal number of total DoFs (100). As the FVM is restricted to one DoF per cell, there are no interior DoFs, increasing the necessary MPI effort. The DGM, on the other hand, has an increasing number of interior DoFs that, for a collocation method, do not contribute to the numerical flux. For the present fourth-order scheme, the ratio of interior to total DoFs is 36%. significant grow with increasing problem size, thus supporting the claim that DG methods are more favorable for highly parallel architectures. A similar trend is visible from the table reporting overall runtimes. The time step restrictions, as discussed, weaken the computational advantage gained in the solution of the semidiscrete system. However, the difference in run time still decreases noticeably with increasing problem and hardware size. For the largest machine with 128 cores, solution times are almost comparable, whereas in the case of the smallest machine, the FV model computed a solution about 42% faster. ### Laser Powder Bed Fusion

As a last example, we will investigate a real multiphysics problem to demonstrate the proposed method for more complex settings. We outline the results from the problem analysis summarized in Figure 5 and indicate how one may implement the resulting numerical scheme. The problem of interest is the fluid flow and temperature field that is present during the laser-based heating of metallic powder. Combined with a moving heat source that travels along a set path and layer-wise re-application of fresh powder, this technique results in the additive manufacturing process of powder bed fusion of metallic materials using a laser beam (PBF-LB/M). Having detailed knowledge about the fluid flow and temperature fields is crucial to obtain dense, that is, pore-free parts with a favorable microstructure. The former is governed by the morphology of the solidified melt pool. The latter is heavily influenced by the spatial and temporal temperature gradients. This specific problem has been tackled many times in the literature using various numerical methods. An encompassing review of the technology as well as recent simulation approaches can be found in [99]. #### 5.3.1 Physics and Governing Equations

Due to the presence of solid, liquid, and gaseous phases, resolving the thermo fluid dynamics of the process involves a multitude of physics.