The role of the small additional terms in the artificial compressibility methods derived from the analysis of the lattice Boltzmann equation

The traditional artificial compressibility method (ACM) directly discretized by the finite difference is challenged by numerical instability resulting in low efficiency. While recent research through analyzing the single-relaxation-time lattice Boltzmann equation (LBE) found that some small additional terms (SATs) are crucial to stabilize the computation and proposed the more actual macroscopic equations (MAMEs). The MAMEs incorporate both advantages of LBE and ACM including good stability and high efficiency, low virtual memory cost respectively. In this paper, we reanalyze the role of the SATs by using the Maxwell iteration, and find that those SATs can offset the second-order temporal derivatives produced by the first-order forward difference in the MAMEs which means the SATs are also of importance to the numerical accuracy. According to this, we propose the modified SATs and apply them to the MAMEs to simulate the two-phase flows using the free energy approach.