Flow in porous media with strong absorption
We solve the so-called porous medium equation with strong absorption
u_t = div ( grad u^m) - u^p,
m>0, 0<p<1, accompanied with zero Dirichlet boundary conditions and some special initial datum.
We simulate numericaly (using the implementation of Jaeger-Kacur scheme) the so-called finite extinction phenomenon of the solution. In the points, where u vanishes (and so the diffusion coefficient m u^(m-1) is equal 0) the governed equation degenerates. The set of such points is called the interface. This degeneracy (of slow diffusion type) causes the finite speed of interface motion. The influence of both degenerate diffusion and strong absorption term u^p makes the dynamics of interfaces complicated and interesting.
Example 1.
Figure1a. Initial profile
Figure1b. Time evolution of the support of the solution; one can see the support shrinking to circle like point in the extinction time.
Example 2.
Figure 2a. Initial profile - halftorus.
Figure 2b. Time evolution of the support; extinction set is a circle.
Example 3.
Figure 3a. Initial profile - perturbed halftorus.
Figure 3b. Time evolution of the support; the extinction set is point.
Related publications:
K.Mikula, Numerical solution of nonlinear diffusion
with finite extinction phenomenon,
Acta Math. Univ. Comenianae
64/2 (1995), 173-184
