## 5 Examples
### Allen-Cahn Equation  
First, we consider the following scalar PDE together with zero flux boundary conditions to be imposed at the four borders of a rectangular domain \(\Omega:[0;L]\times[0;W]\):  
\[\frac{1}{K}\partial_{t}\phi-\Delta\phi=-\frac{2}{\xi^{2}}\partial_ {\phi}g(\phi)-\frac{\mu_{0}}{3\gamma\xi}\partial_{\phi}h(\phi)\quad\forall\phi \in\Omega, \tag{18}\] \[\frac{\partial\phi}{\partial n}=0\quad\forall x\in\partial\Omega. \tag{19}\]  
This equation is called the Allen-Cahn equation and describes the time-evolution of a scalar, nonconserved order-parameter field \(\phi\), as is often called the phase field. The equation is commonly used in the modeling of self-organized microstructure evolution or complex pattern formation processes, as driven by local thermodynamics and/or mechanics. The phase field variable \(\phi\) can be understood as a coloring function that locally indicates the presence or absence of a certain phase or a certain material state within a given microstructure. For instance, in modeling of microstructure evolution during solidification, \(\phi=1\) may denote the local presence of the solid and \(\phi=0\) may denote the local presence of the liquid phase [61; 62]. If applied to the description of crack propagation, the order-parameter field \(\phi\) is understood as the local material state, which can be either broken \(\phi=1\) or not \(\phi=0\)[69; 70]. The scalar quantities \(K\), \(\bar{\xi}\), \(\mu_{0}\), and \(\gamma\) are model constants that determine the evolution of the scalar field \(\phi\), and we adopt the notation of [71]. The polynomials \(g\) and \(h\) on the right-hand side of Equation (18) pose a nonlinearity to the equation. Their derivatives are given by \(\partial_{\phi}g(\phi)=2\phi(1-\phi)(1-2\phi)\) and \(\partial_{\phi}h(\phi)=\delta\phi(1-\phi)\). In the following, we will gather those polynomial terms in the joint potential term \(f(\phi)=2\partial_{\phi}g(\phi)/\xi^{2}+\mu_{0}\partial_{\phi}h(\phi)/3\gamma\xi\). Further details on the parametrization of the model are given in the Appendix A Table A. Concerning the Allen-Cahn equation, we will consider two different scenarios, highlighting different aspects of the physics behind the equation. For each of these two scenarios, we formulate quantitative measures to be able to adequately question the accuracy of the numerical solutions. In the first scenario, we consider the motion of a planar interface between two phases at different energy density levels. The low-energy phase is expected to grow at the expense of the high-energy phase, which induces a motion of the interface between them at a velocity proportional to the constant energy density difference \(\mu_{0}\). The scenario is realized as a quasi 1D problem \([0;L=100]\times[0;W=1]\), where the interface normal direction is pointing in the x-direction and the use of simple von Neumann boundary conditions with zero phase field fluxes at the borders of the rectangular domain is legitimate. The realization of this scenario with tilted interface orientations, including the formulation of appropriate boundary conditions on the borders of the rectangular domain, is discussed in detail in [62]. In this highly symmetric quasi-1D case, the scenario can be quantitatively evaluated using the existing analytic solution for the phase field:  
\[\phi(x,t)=\frac{1}{2}\bigg{(}1-\tanh\frac{x-\bar{x}(t)}{\bar{\xi}}\bigg{)}, \tag{20}\]  
where the time dependence of the central interface position \(\bar{x}\) is given by \(\bar{x}(t)=x_{0}+M\mu_{0}t/\gamma\), with the initial position at \(x_{0}=20\). The initial condition of this problem is, thus, formed by evaluating Equation (20) at time zero. To investigate the impact of arithmetic complexity on computational efficiency, we will seek an approximation of this real-valued function \(\phi(x,t)\) in one spatial dimension on a \([0;100]\) grid with equispaced vertices.