We study the asymptotical compatibility of the Fourier spectral method in multidimensional space for the Nonlocal Ohta-Kawasaka (NOK) model, which is proposed in our previous work. By introducing the Fourier collocation discretization for the spatial variable, we show that the asymptotical compatibility holds in 2D and 3D over a periodic domain. For the temporal discretization, we adopt the second-order backward differentiation formula (BDF) method. We prove that for certain nonlocal kernels, the proposed time discretization schemes inherit the energy dissipation law. In the numerical experiments, we verify the asymptotical compatibility, the second-order temporal convergence rate, and the energy stability of the proposed schemes. More importantly, we discover a novel square lattice pattern when certain nonlocal kernel are applied in the model. In addition, our numerical experiments confirm the existence of an upper bound for the optimal number of bubbles in 2D for some specific nonlocal kernels. Finally, we numerically explore the promotion/demotion effect induced by the nonlocal horizon, which is consistent with the theoretical studies presented in our earlier work.

翻译：本文研究了多维空间中非局部Ohta-Kawasaki（NOK）模型（该模型由我们前期工作提出）的傅里叶谱方法渐近相容性。通过引入空间变量的傅里叶配置离散化，我们证明了在周期域上的二维和三维情形下渐近相容性成立。在时间离散方面，我们采用二阶后向差分公式（BDF）方法，并证明对于特定非局部核函数，所提出的时间离散格式保持能量耗散律。数值实验验证了所提格式的渐近相容性、二阶时间收敛率以及能量稳定性。更重要的是，我们发现当模型采用某些非局部核函数时会产生新颖的方晶格图案。此外，数值实验证实了在特定非局部核函数作用下二维空间最优气泡数目存在上界。最后，我们通过数值方法探究了非局部作用域所诱发的促进/抑制效应，这一结果与我们前期工作中的理论研究相吻合。