Аннотация:Numerical scheme for Laplacian growth models.
LOHEAC J.-P. (Jean-Pierre.Loheac@ec-lyon.fr, Ecole centrale de Lyon, France)
This work is common with A.S. Demidov (Moscow State University), it concerns a numer-
ical scheme involved by the Helmholtz-Kirchhoff method applied in the case of Hele-Shaw
flows.
This method allows to transform a free boundary bi-dimensional problem in a fixed bound-
ary problem by introducing a convenient parameterization. It also leads to build numerical
schemes.
For instance, a model of Hele-Shaw flows with punctual source is the Stokes-Leibenson
problem: let
0 be a bi-dimensional bounded simply connected domain such that its
boundary ����0 is smooth enough. This domain will be deformed according to the following
law: at time t, we obtain a domain
t =
of boundary ����t = ���� such that the normal
velocity of each point s 2 ���� is given by the following kinetic condition,
˙s. = @u , (1)
where u is the solution of the following Laplace problem,
u = q , in
, and u = 0 , on ���� . (2)
The Helmholtz-Kirchhoff method leads us to write the problem in the form of a Cauchy
problem for some integro-differential. This can give existence and uniqueness results,
which are global in time when q > 0.
Furthermore, by introducing an approximate Stokes-Leibenson problem concerning polyg-
onal domains, this integro-differential can be expressed in the form of a non-linear differ-
ential equation in a finite-dimension space.
Numerical simulations will be presented and discussed.
Especially some numerical experiments show some critical manifold which can explain
some phemonenon of instabilities.
References
[1] Almgren R., Crystalline Saffman-Taylor fingers, SIAM J. Appl. Math. Math. J. 55 (1995), 1511–
1535.
[2] Demidov A.S., Some applications of the Helmholtz-Kirchhoff method (equilibrium plasma in tokamaks,
Hele-Shaw flow, and high-frequency asymptotics), Russian Journal of Mathematical Physics, 7 (2000),
No. 2, 166–186.
[3] Demidov A.S., Loh´eac J.-P., Numerical scheme for Laplacian growth models based on the Helmoltz-
Kirchhoff method, Analysis and Mathematical Physics. Trends in Mathematics (2009), 107–114.