## 5 Examples
### Allen-Cahn Equation
will be shown, such an additional term can be omitted if the problem is solved using the recommended DG scheme due to the considerably lowered numerical diffusion. In summary, choosing the FV over the recommended DG method, in this case, would not only require evaluating a fully nonlinear variant of the advection equation to gain accurate results, but it would also force the application expert to pick an appropriate value for an artificial model constant, which can be subject to tedious calibration work. In the following, we will compare this particular choice of method with the finite volume method, which would be the next alternative and is, in principle, also well suited to tackle such problems. The continuous Galerkin and finite difference methods do not lend themselves well to solving such equations and will thus be omitted from this benchmark. In particular, the CG method is known to be unstable for first-order hyperbolic equations, as the stability of the scheme can be shown to be dependent on mesh size [42]. One must add that in principle, the finite difference method could be applied here, where, however, two different limitations apply. First, the only choice of stencil that would be stable for this equation is the forward difference (or upwind) approximation. This choice, however, is not covered by the proposed decision process, as it is formally equivalent to a continuous Petrov Galerkin method. One can show that this scheme corresponds to a simplified streamline upwind Petrov Galerkin (SUPG) method. As we have restricted ourselves for the sake of decidability to Bubnov Galerkin methods, this stencil is not admissible here. Secondly, using the finite difference method here implies the strict use of a Cartesian grid. We thus proceed to write the weak form for Equation (26) by multiplying with a discrete test function \(v_{h}\), integrating over the whole domain \(\Omega\) and subsequently performing integration by parts. Find \(\alpha_{h}\in V_{h}\) such that

\[\int_{\Omega}v_{h}\frac{\partial\alpha_{h}}{\partial t}\,dx+\int _{\Gamma}v_{h}\alpha_{h}(\underline{u}\cdot\underline{u})\,dS-\int_{\Omega}( \
abla v_{h}\cdot\underline{u})\alpha_{h}\,dx=0,\qquad\forall v_{h}\in V_{h}, \tag{32}\] \[\alpha_{h,\tilde{\Gamma}_{h,j}}=\alpha_{h,\tilde{\Gamma}_{j}, \sigma},\quad\forall x\in\partial\Omega, \tag{33}\]

where \(\Gamma\) denotes the union of all interior and exterior facets of the domain and \(\tilde{\Gamma}\) are the subsets of the domain boundary \(\partial\Omega\). In this case, specifically, \(\tilde{\Gamma}_{b,l}\) are the slave facets at the bottom and left boundary that the values of the slave facets from the top and right master facets \(\tilde{\Gamma}_{t,r}\) are mapped to. This PDE in combination with periodic boundaries possesses an analytical solution of the form

\[\alpha(x,t)=\alpha(x-ut,0). \tag{34}\]

That is, after traversing the quadratic domain with the given velocity \(\underline{u}= \begin{bmatrix}1&1\end{bmatrix} ^{T}\), the solution field must exactly correspond to the initial condition. Verification of numerical results is, thus, very straightforward. We solve this problem using the Firedrake problem-solving environment along with the popular libraries PETSc and Scotch for efficient parallel computing [3, 88, 89, 90, 91, 92, 93, 94, 95]. As the finite volume method can simply be understood as a discontinuous Galerkin method of polynomial degree zero, the implementation is virtually the same for both schemes. Note that for the finite volume method, the last term in Equation (32) becomes zero since the derivative of a constant vanishes. Thus, we omit this term from the assembly to save computations and to more accurately represent the arithmetic intensity posed by the original formulation of this scheme. For the sake of visualization, we project the solution onto a first-degree space with \(H^{1}\) continuity. The equations are solved by a three-stage implicit Runge-Kutta method. In both cases, careful attention has to be paid regarding the time step.