Fast and robust solvers for pressure-correction in bubbly flow problems

We consider the numerical simulation of two-phase fluid flow, where bubbles or droplets of one phase move against a background of the other phase. Such flows are governed by the Navier–Stokes equations, the solution of which may be approximated using a pressure-correction approach. Within such an approach, the computational cost is often dominated by the solution of a linear system corresponding to a discrete Poisson equation with discontinuous coefficients. In this paper, we explore the efficient solution of these linear systems using robust multilevel solvers, such as deflated variants of the preconditioned conjugate gradient method, or robust multigrid techniques. We consider these families of methods in more detail and compare their performance in the simulation of bubbly flows. Some of these methods turn out to be very effective and reduce the amount of work to solve the pressure-correction system substantially, resulting in efficient calculations for two-phase flows on highly resolved grids.