Completing a well-balanced numerical method for a model of two-phase flows by computing correctors

We complete a well-balanced numerical method by introducing computing correctors to an earlier scheme for a model of two-phase flows. Each improvement based on a corrector to the scheme is designed to reduce the size of the errors across the interface of each node when using the solid contact to absorb the nonconservative terms. Three correctors of two kinds are presented. One corrector of the first kind is designed to correct the states on both side of the solid contact at each node and the corresponding numerical flux before applying the iterative scheme. Two correctors of the second kind are designed to correct the state given by the iterative scheme depending on the sign of the velocity of the solid contact. These improvements are still well-balanced schemes. Tests show that the improvement by using the corrector of the first kind gives relatively better results, and the improvements by using one corrector of the second kind give much better results. Interestingly, we find that improvements by using a corrector of second kind can resolve the accuracy problem of the existing scheme when its approximate solutions might converge to the solution slightly different from the exact solution.