Numerical Approaches for Porous Media Flows
In this talk we discuss some numerical methods that are appropriate
for models describing porous media flows. Such problems are typically
parabolic nonlinear, but - depending on the values of the unknown -
may degenerate into elliptic or hyperbolic ones. As a consequence, the
solution may lack regularity and develop singularities.
Appropriate numerical algorithms are requested for dealing with such
kind of problems. We focus here on the particular case of a scalar
model that can be formulated in a fixed domain. To overcome the
difficulties that are due to the degeneracy, we first perturb the
original model and end up with a regular parabolic one. For the
resulting we employ standard discretization methods (implicit in time
and an upwinded box scheme in space) and show that these lead to
stable and convergent algorithms. A few words will be said on how to
solve the time discrete nonlinear problems.