Investigation into a Broyden-Type method for the Fully-Implicit scheme of Two-Fluid model and its convergence performance

The fully-implicit scheme of two-fluid model enables calculation with large time step, and it is attractive for the long problem time simulation and calculation with fine cells. One of the key aspects of the solution algorithm is the calculation of Jacobian matrix. First, the Jacobian matrix of fully-implicit scheme is ill-conditioned. We find that non-dimensionalizing the governing equations and primary variables can reduce the condition number effectively. Second, computing the Jacobian matrix of the fully-implicit scheme is time-consuming. To address this issue, we adopt Broyden-Schubert method. This method is not only easily implementable and has a fast computational speed but also can maintain the sparse structure of the matrix. However, it leads to smaller convergence region and lower convergence rate. A very natural idea is to calculate Jacobian matrix using direct calculation method for the first few steps, and then employ Broyden-Schubert method for the remaining steps. It is found that a small number of direct calculation iterations can significantly improve convergence performance. Therefore, this hybrid method may be a potential development direction for the fully-implicit scheme. It is important to note that the conclusions presented in this paper are derived from near-steady-state and relatively simple transient cases. As a result, the applicability of the Broyden-type method to complex two-phase flow transient cases cannot be guaranteed. Therefore, further test of its applicability in two-phase flow is essential and needs to be conducted in future research work.