High-order analysis of lattice Boltzmann models for the conservative Allen-Cahn equation

The conservative Allen-Cahn equation has been widely used to capture complicated interfacial problems. As a typical convective-diffusion equation, it can be solved within the framework of the lattice Boltzmann (LB) method. In this work, we focus on three modified LB schemes with a temporal or spatial difference-based correction term. The first one is proposed by our present work, while the other two are presented for comparative studies. All of them are able to recover the Allen-Cahn equation accurately up to the second order. Through a high-order truncation analysis, the dominant error terms of these modified LB schemes are extracted at the third and fourth order for the first time. Comparative studies among different models are performed in terms of accuracy, convergence rate, boundedness, and efficiency. Our proposed model shows superiority in reducing numerical errors and maintaining boundedness over a wide range of relaxation time, which is consistent with our theoretical analysis. In addition, we improve the locality of our modified model by reorganizing the collision step and adding a correction step after the streaming process.