@TechReport{ it:2012-019,
author = {Owe Axelsson and Xin He and Maya Neytcheva},
title = {Numerical Solution of the Time-Dependent {N}avier-{S}tokes
Equation for Variable Density--Variable Viscosity},
institution = {Department of Information Technology, Uppsala University},
department = {Division of Scientific Computing},
year = {2012},
number = {2012-019},
month = aug,
abstract = {We consider methods for numerical simulations of variable
density incompressible fluids, modelled by the
Navier-Stokes equations. Variable density problems arise,
for instance, in interfaces between fluids of different
densities in multiphase flows such as appear in porous
media problems. It is shown that by solving the
Navier-Stokes equation for the momentum variable instead of
the velocity, the corresponding saddle point problem, which
arises at each time step, becomes automatically
regularized, enabling elimination of the pressure variable
and leading to a, for the iterative solution, efficient
preconditioning of the arising block matrix. We present
also stability bounds and a second order operator splitting
method. The theory is illustrated by numerical experiments.
For reasons of comparison we also include test results for
a method, based on coupling of the Navier-Stokes equations
with a phase-field model.}
}