A positivity preserving all-speed projection method for simulating multiphase flows is presented. There are two main improvements that enable the present method to outperform the following recent work reported by Jemison, Sussman and Arienti (Journal of Computational Physics, volume279, 2014): (i) momentum is stored and updated on the MAC grid instead of on the collocated cell center grid, this eliminates the need for interpolating pressure to the MAC grid from the cell centers, and (ii) instead of solving for the total energy for each material, we formulate our numerical algorithm so that we solve for the entropy for each material, this enables our method to be positivity preserving for both density and internal energy. In the same spirit as the recent 2014 work (Jemison, Sussman, Arienti), a unified method for simulating multiphase flows using an exactly mass, momentum, and entropy conserving Cell-Integrated Semi-Lagrangian advection algorithm is presented. The deforming material boundaries are represented using the moment-of-fluid method. The new algorithm uses a semi-implicit pressure update scheme that asymptotically preserves the standard incompressible pressure projection method in the limit of infinite sound speed. The asymptotically preserving attribute makes the new method applicable to compressible and incompressible flows including stiff materials; enabling large time steps characteristic of incompressible flow algorithms rather than the small time steps required by explicit methods. Shocks are captured and material discontinuities are tracked, without the aid of any approximate or exact Riemann solvers. Simulations of underwater explosions and fluid jetting in one, two, and three dimensions are presented which illustrate the effectiveness of the new algorithm at efficiently computing multiphase flows containing shock waves and material discontinuities with large \impedance mismatch." This is research with M. Vahab, M. Arienti, M. Jemison, and M.Y. Hussaini.

Mark Sussman was born in San Diego, California in 1966. He graduated from San Diego State University in 1986 (Bachelor's degree in Applied Mathematics, minor in physics and computer science), and University of California, Los Angeles in 1994 (PhD degree in mathematics, advisor S. Osher). Mark Sussman's employment history includes two years as a postdoc at Lawrence Livermore/Berkeley National Laboratory, three years as a visiting research assistant professor at University of California Davis, and 16 years at Florida State University (now a professor in the Mathematics department). Mark Sussman's research specializes in numerical methods for deforming boundary problems with applications to multiphase flow. Mark Sussman has authored/co-authored many articles in refereed archival journals, served as a referee for articles submitted to archival journals in the fields of mathematics, engineering, biology, chemistry, and computer science, and has served on many proposal review panels.