## 3 Computational Tests  
This section presents a numerical investigation of the coarsening dynamics under various initial conditions and domains. First, let us consider the following initial condition for diblock copolymers:  
\[\phi(x,y,0)=0.25\text{rand}(x,y)+\hat{\phi}(x,y), \tag{16}\] \[\text{where}\ \hat{\phi}(x,y)= \begin{cases}-0.3,&\text{if}\ \ \sqrt{(x-7.5)^{2}+(y-7.5)^{2}}>5,\\ 0,&\text{otherwise}.\end{cases}  \tag{17}\]  
Here, \(\text{rand}(x,y)\) denotes a random number between \(-1\) and \(1\) and \((x,y)\in[0,15]\times[0,15]\). In this experiment, we fix the initial condition and all parameters except for the spatial resolution to investigate its influence on the solution. Specifically, we set \(h=L_{x}/N_{x}\), \(\epsilon=0.03\), \(\sigma=100\), and \(\Delta t=0.1\). As shown in Figure 1, we choose three spatial resolutions--\(N_{x}=N_{y}=200\), \(300\), and \(400\)--and examine the solution at time \(t=12\),\(000\Delta t\). The differences are clearly visible: when the spatial resolution is low, as in Figure 1a, the solution appears non-smooth. As the spatial resolution increases, the solution becomes noticeably smoother and more regular. Figure 1: (a–c) are the snapshots of numerical solutions of the nonlocal CH model (4) with the initial condition (18) using \(N_{x}=N_{y}=200\), \(N_{x}=N_{y}=300\), and \(N_{x}=N_{y}=400\), respectively, at time \(t=12,000\Delta t\). Then, let us consider another initial condition for diblock copolymers:  
\[\phi(x,y,0)=0.25\text{rand}(x,y)+\tilde{\phi}(x,y), \tag{18}\] \[\text{where}\ \tilde{\phi}(x,y)= \begin{cases}-0.3,&\text{ if }\ \sqrt{(x-7.5)^{2}+(y-7.5)^{2}}\leq 5,\\ 0,&\text{ otherwise.}\end{cases}  \tag{19}\]  
Figure 2 shows the numerical results at different evolution states. We use \(N_{x}=N_{y}=500\), \(h=0.03\), \(\epsilon=h\), and \(\Delta t=0.1\). The computational results show well the coarsening dynamics of the nonlocal CH equation. Figure 2 shows the temporal evolution of the phase field variable \(\phi\), where the left snapshot corresponds to \(t=100\Delta t\) while the right snapshot to \(t=2000\Delta t\). Initially, a disordered structure with high-frequency oscillations appears within a circular region as a result of the random perturbation that was applied to the central domain. Over time, as shown at time \(t=100\Delta t\), small-scale patterns begin to merge and form more coherent stripe-like morphologies. Eventually, at time \(t=2000\Delta t\), the system exhibits a significant reduction in the number of interfaces and forms well-defined, large-scale labyrinthine structures. The circular boundary remains intact throughout the process, due to the imposed initial configuration \(\tilde{\phi}\), which introduces a contrast between the inside and outside regions. These results effectively capture the phase separation and coarsening dynamics of the diblock copolymer system in regions characterized by different average values of the phase variable. We define the discrete total energy as  
\[\mathcal{E}^{h}(\phi^{k})=\sum_{m=1}^{N_{x}}\sum_{n=1}^{N_{y}}F( \phi^{k}_{mn})h^{2} + \frac{h^{2}}{N_{x}N_{y}}\sum_{p=-N_{x}/2+1}^{N_{x}/2}\sum_{q=-N_{y }/2+1}^{N_{y}/2}\frac{e^{2}}{2}(|\mathcal{E}_{p}\hat{\phi}^{k}_{pq}|^{2}+| \eta_{q}\hat{\phi}^{k}_{pq}|^{2}) \tag{20}\] \[+ \frac{h^{2}}{N_{x}N_{y}}\sum_{p=-N_{x}/2+1}^{N_{x}/2}\sum_{q=-N_{y }/2+1}^{N_{y}/2}\frac{\sigma}{2}(|\mathcal{E}_{p}\hat{\psi}^{k}_{pq}|^{2}+| \eta_{q}\hat{\psi}^{k}_{pq}|^{2}),\]  
where \(\psi\) is obtained by solving \(-\Delta\psi=\phi-\tilde{\phi}\). Figure 2 also shows the time evolution of the normalized discrete total energy \(\mathcal{E}^{h}(\phi^{k})/\mathcal{E}^{h}(\phi^{0})\). The normalized discrete total energy initially decreases over time and gradually stabilizes afterward. Figure 2: Time evolution of the normalized discrete total energy \(\mathcal{E}^{h}(\phi^{k})/\mathcal{E}^{h}(\phi^{0})\). Furthermore, we present the phase separation behavior with another initial condition via a nonlocal CH model for diblock copolymers.