A second order primitive preconditioner for solving all speed multi-phase flows

In this paper, we present a new second order primitive preconditioner technique for solving all speed multi-phase flow problems. With this technique, one can compute both compressible and incompressible flows with Mach-uniform accuracy and efficiency (i.e., accuracy and efficiency of the method are independent of Mach number). The new primitive preconditioner can handle both strong and weak shocks, providing highly resolved shock solutions together with correct shock speeds. In addition, the new technique performs very well at the zero Mach limit. In the case of multi-phase flow, the new primitive preconditioner technique enables one to accurately treat phase boundaries in which there is a large impedance mismatch. The present method is tested on a variety of problems from low (low speed) to high Mach number (high speed) flows including multi-phase flow tests, i.e., computing the growth and collapse of adiabatic bubbles for study of underwater explosions. The numerical results show that the newly proposed method supersedes existing up-to-date numerical techniques in its category.