Monolithic Newton-Multigrid Solver for Multiphase Flow Problems with Surface Tension

We have developed a monolithic Newton-multigrid solver for multiphase flow problems which solves velocity, pressure and interface position simultaneously. The main idea of our work is based on the formulations discussed in [1], where it points out the feasibility of a fully implicit monolithic solver for multiphase flow problems via two formulations, a curvature-free level set approach and a curvature-free cutoff material function approach. Both formulations are fully implicit and have the advantages of requiring less regularity, since neither normals nor curvature are explicitly calculated, and no capillary time restriction. Furthermore, standard Navier-Stokes solvers might be used, which do not have to take into account inhomogeneous force terms. The reinitialization issue is integrated with a nonlinear terms within the formulations.The nonlinearity is treated with a Newton-type solver with divided difference evaluation of the Jacobian matrices. The resulting linearized system inside of the outer Newton solver is a typical saddle point problem which is solved using the geometrical multigrid with Vanka-like smoother using higher order stable FEM pair $Q_2/P^{\text{disc}}_1$ for velocity and pressure and $Q_2$ for all other variables. The method is implemented into an existing software packages for the numerical simulation of multiphase flows (FeatFlow). The robustness and accuracy of this solver is tested for two different test cases, i.e. static bubble and oscillating bubble, respectively [2].REFERENCES[1] Ouazzi, A., Turek, S. and Damanik, H. A curvature-free multiphase flow solver via surface stress-based formulation. Int. J. Num. Meth. Fluids., Vol. 88, pp. 18–31, (2018).[2] Afaq, M. A., Turek, S., Ouazzi, A. and Fatima, A. Monolithic Newton-Multigrid Solver for Multiphase Flow Problems with Surface Tension. Ergebnisberichte des Instituts fuer Angewandte Mathematik Nummer 636, Fakultaet fuer Mathematik, TU Dortmund University, 636, 2021.