Reconstructed small additional terms for the problem of vapor–liquid phase change

The computation of the traditional artificial compressibility method (ACM) applying explicit finite difference scheme is very efficient but challenged by numerical instability. Compared to it, the lattice Boltzmann method is more stable, and the mechanism of good stability was recently explored by Lu et al. through analyzing single-relaxation-time lattice Boltzmann equation (LBE). They found that some small additional terms (SATs) are crucial to stabilize the computation and proposed the more actual macroscopic equations (MAMEs). The MAMEs combine both advantages of the LBE and ACM like good stability, high efficiency, and low virtual memory cost. Although they claimed those small terms had no obvious effect on the accuracy of numerical solutions, the original SATs cannot give a precise prediction of coexisting densities for the problem of vapor–liquid phase change. In this paper, we reanalyze the role of the SATs by using the Maxwell iteration and find 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. Based on this analysis, reconstructed SATs according to the Maxwell iteration and a finite difference solver are proposed for the system of vapor–liquid phase change. Numerical results validate the reconstructed SATs and the present solver.