A Fully Implicit, Conservative, Front Tracking Method for All-Speed Multi-fluid Flows

A novel front-tracking method is developed for simulation of compressible all-speed multi-fluid flows. The prominent features of our method are: $a)$ it is \textit{fully implicit}, based on Jacobian-Free Newton-Krylov (JFNK) framework, opening the possibility to efficiently simulate multi-physics problems with a wide spread in time scales; $b)$ it is \textit{fully conservative}, even near the contacts, without any adverse consequences with pressure/velocity oscillations; $c)$ hybridization of the Eulerian treatment in the bulk-fluid with the semi-Lagrangian treatments of cut-cells near multi-material interfaces enables \textit{efficient high-order-accurate} \textit{spatial discretization}, capturing interfacial jumps sharply, within one cell; $d)$ \textit{interfacial geometry} \textit{is a part of the JFNK solution vector}, avoiding operator-split treatment of the interface and near-interfacial fluid flows, which is a state-of-the-art in all previous (explicit) methods for interface tracking. We discuss and demonstrate the effectiveness of the Implicit Continuous-fluid Eulerian (ICE) physics-based preconditioner (PBP) of the Krylov (GMRES) method used as a linear solver in our Newton-based implicit Runge-Kutta time discretization. It will be shown that the ICE-PBP collapses all generally-complex eigenvalues of the Jacobian matrix to the real axes, effectively clustering eigenvalues and thus enabling a significant speed-up of convergence of the linear solver.