Analysis of time integration methods for the compressible two-fluid model for pipe flow simulations

•BDF2 is proposed as time integration method for the two-fluid model.•Accuracy and stability are investigated via eigenvalue analysis of the continuous, semi-discrete and fully-discrete equations.•A novel automated von Neumann analysis is developed, which does not require symbolic manipulations or source code knowledge.•Discrete Flow Pattern Maps are used to determine if the discrete well-posed unstable flow regime matches the theoretical one.•Starting from well-posed unstable conditions, the two-fluid model often turns ill-posed before slugs are formed.