## 3 Computational Tests
The initial condition is given by  
\[\phi(x,y,0)=0.25\text{rand}(x,y)+\tilde{\phi}(x,y), \tag{21}\] \[\text{where }\tilde{\phi}(x,y)= \begin{cases}-0.3,&\text{if }\ 3\leq \sqrt{(x-7.5)^{2}+(y-7.5)^{2}}\leq 6,\\ 0,&\text{otherwise}.\end{cases}  \tag{22}\]  
A computational domain is given as \([0,15]\times[0,15]\). Here, we use \(N_{x}=N_{y}=500\), \(h=0.03\), \(\sigma=105\), \(\epsilon=h\), and \(\Delta t=0.1\). The final time is \(T=800\). Figure 3 shows the phase separation behaviors during evolution and illustrates the time evolution of the phase field governed by the nonlocal CH equation under a distinct annular initial configuration. The initial condition introduces a negative average phase value only in a ring-shaped region between two concentric circles. At the early stage, the phase field would exhibit a largely homogeneous pattern outside the ring, while fine-grained oscillatory structures emerge within the annular region. These features stem from the imposed random perturbation and the localized nonzero mean in the initial configuration. As time progresses (\(t=100\Delta t\), the left snapshot in the Figure 3), the internal structure within the ring undergoes coarsening and results in a hexagonal dot pattern, whereas stripe-like morphologies appear in the surrounding region. This contrast in the evolved patterns reflects the effect of the spatial variation in the initial average phase distribution. Eventually, in the late stage (\(t=2000\Delta t\), the right snapshot in the Figure 3), a well-organized pattern with a hexagonal dot structure appears inside the annular region, while elongated lamellae with clearly defined interfaces emerge in the other regions. These results demonstrate that spatially selective initial perturbations can effectively localize phase separation and pattern formation. This localization induces different morphological evolution within diblock copolymer systems that are described by the nonlocal CH equation. Figure 3 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. Another initial condition is given as the shape of a zebra. \[\phi(x,y,0)=0.25\text{rand}(x,y)+\tilde{\phi}, \tag{23}\]  
Figure 3: Time evolution of the normalized discrete total energy \(\mathcal{E}^{h}(\phi^{k})/\mathcal{E}^{h}(\phi^{0})\). \[\text{where}\ \tilde{\phi}= \begin{cases}0,&\text{if}\ \ \ (x,y)\in\Omega_{\text{ zebra}},\\ -0.25,&\text{otherwise}.\end{cases}  \tag{24}\]  
Here, \(\Omega_{zebra}\) is defined as the domain of the shape of the zebra in this particular experiment. A computational domain is given as \([0,29]\times[0,22]\). Parameter values are set to \(N_{x}=580\), \(N_{y}=440\), \(h=0.1\), \(\sigma=53\), \(\epsilon=h\), and \(\Delta t=0.001\). The final time is \(T=10\). Figure 4 describes the phase separation behaviors over time and illustrates the temporal evolution of phase separation governed by the nonlocal CH equation with an initial condition shaped like a zebra. In this setup, the phase variable \(\phi\) is initialized to random values with an average \(\tilde{\phi}=0\) within the zebra-shaped domain \(\Omega_{\text{zebra}}\), while the region outside the shape is set to random values with an average \(\tilde{\phi}=-0.25\). Initially, a disordered mixture of short stripe and dot-like patterns would emerge within the zebra-shaped region. The interfaces are irregular and the pattern has not yet coarsened significantly. As time progresses to \(t=1000\Delta t\) (the left snapshot in the Figure 4), the internal structure becomes more organized, with the appearance of elongated stripe-like morphologies that begin to align spatially. The boundaries of the zebra remain sharply defined, which indicates that the pattern formation is effectively confined to the initial domain.