Asymptotically Compatible Schemes for Nonlocal Ohta Kawasaki Model

TL;DR

This paper develops and verifies asymptotically compatible Fourier spectral schemes for the Nonlocal Ohta-Kawasaki model in multiple dimensions, demonstrating energy stability, convergence, and discovering novel patterns and bounds through numerical experiments.

Contribution

It introduces a Fourier spectral method with proven asymptotic compatibility and energy stability for the NOK model, including new pattern discovery and bounds on bubble numbers.

Findings

Verification of asymptotical compatibility in 2D and 3D

Second-order temporal convergence demonstrated

Discovery of a novel square lattice pattern

Abstract

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.