## 5 Examples
### Allen-Cahn Equation
For hyperbolic problems of such time, the time step where stability is given is strictly bounded by the Courant-Friedrichs-Lewy number \(\text{CFL}\leq\frac{1}{2k+1}\) for a scheme of degree \(k\)[96]. One can easily verify that for a finite volume scheme, this corresponds to the well-known condition that the CFL number must stay at or below unity. Not only does that mean, regarding arithmetic complexity, that the FVM has a simplified assembly process, but it also means that the admissible time step is, in general, larger than for DG methods. This discrepancy drastically increases with the polynomial order taken for the DGM. The corresponding linear systems are each solved using the same matrix-free algorithm with Krylov subspace preconditioning. The corresponding results of the simulation are shown in Figure 9.

The differences in accuracy are obvious. The DG method can capture the rectangular profile throughout the simulation with relatively good accuracy, while the FV simulation is strongly diffused. The latter is due to the missing artificial interface compression, given in Equation (30), resulting in a substantial numerical loss of the conserved quantity \(\alpha\). We note at this point that there are formulations of the FVM that capture shocks much more accurately whilst controlling oscillations. Due to the maturity of this method, the field of constructing weighted essentially nonoscillatory (WENO) schemes is quite advanced and, also, in this case, will yield better approximations. Such schemes are also applicable to considerably more complex systems of equations [97]. However, this argument also applies to the DGM since it is, as has been shown, an extension of the FVM. To make the comparison fair and to be able to rely on existing tools, we chose to compare both schemes with the same, relatively simple reconstruction technique. For both cases, the use of WENO schemes would increase the computational load considerably, as these need to reconstruct polynomial approximations for each flux using relatively wide local stencils--similar to a high-order finite difference scheme [36]. The choice of higher-order reconstruction techniques, however, should not affect the qualitative difference in approximation properties and performance. For example, [PERSON] et al. reported very similar findings for higher-order WENO schemes for the advection equation in two dimensions [9]. In addition to the previous comparison regarding accuracy, we also benchmark the capabilities for parallel computing using the three machines given in Table 4. To properly scale up this problem, we aim to keep the number of degrees of freedom per core constant for the three environments, that is, in this case around 18.400 DoFs/core. This is in the vicinity of the 25.000 DoFs/core regime, where beyond that point, considerable drop-offs in performance are to be expected [98]. According to the theory, the DGM should perform with a noticeably higher efficiency due to less communication overhead between processes. This is due to the reduced number of cells for the same number of DoFs; as such, the DGM has a much higher ratio of DoFs that lie inside the cell instead of at the boundary. As a consequence, there are fewer DoFs in relative terms that require the evaluation of a

Figure 9: Solution of the two-dimensional advection equation using the finite volume and discontinuous Galerkin method. Both models were run using the identical amount of global degrees of freedom set to 147.456. Element facets are drawn in white to illustrate the difference in grid size. (**a**) Finite volume solution at end time on a 384 \(\times\) 384 grid. The solution is heavily diffused to a parabolic profile. (**b**) Discontinuous Galerkin solution at end time on a 96 \(\times\) 96 grid using third-order polynomials. One can observe the presence of spurious oscillations that remain stable in magnitude throughout the simulation. numerical flux and, thus, not as much overhead due to MPI efforts.