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©2012 Civil-Comp Ltd |
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K. Tomoeda
Department of Applied Mathematics and Informatics, Osaka Institute of Technology, Japan
Keywords: nonlinear diffusion, free boundary, interface, support splitting, support merging, difference scheme.
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reference
We are concerned with the dynamical behaviour of non-stationary seepage in a
non-linear filtration. The representative filtration is well known as the flow
through porous media where the water evaporates. In particular, it is expected that
such a seepage exhibits support splitting and merging phenomena, which are caused
by the interaction between the nonlinear diffusion and the penetration of the fluid
from the boundary on which the flowing tide and the ebbing tide occur. Here the
support means the region where the fluid exists.
To model such phenomena in one dimensional space we introduce a model based on
the equation described as the nonlinear initial boundary value problem, which is
used to describe the flow through porous media with absorption [1,2]. This equation
is also used to describe the propagation of thermal waves in plasma physics [3].
From analytical points of view, the existence and uniqueness of a weak solution
and the comparison theorem are proved by Bertsch [4]. For the initial-boundary
value problem Kersner [5] proved the appearance of support splitting phenomena,
but he did not show that support merging phenomena appear after the support splits.
To investigate such phenomena to this initial boundary value problem it is important
to construct a numerical method to it and to analyze the profile of the support of the
stationary solution of it. We obtain the following results:
- Numerical solutions given by our difference scheme converge to the exact one as the
space mesh width tends to zero;
- A stationary solution of the initial boundary value problem exists and is unique.
- Stabilization Theorem; that is, the solution of the initial boundary value problem
converges to a stationary solution as time tends to the infinity.
The proof of 1) can be obtained by the similar argument as stated in [6]. We also
prove 2) by use of the standard theory of ordinary differential equations. The
stabilization theorem is proved by use of the omega-limit theory and some inequality
given by Bertsch [4].
We demonstrate some interesting numerical solutions, which show support splitting and
merging phenomena.
- 1
- P.Y. Polubarinova-Kochina, "Theory of Ground Water Movement", Princeton Univ. Press, 1962.
- 2
- A.E. Scheidegger, "The Physics of Flow through Porous Media", Third edition, University of Toronto Press, 1974.
- 3
- P. Rosenau, S. Kamin, "Thermal waves in an absorbing and convecting medium", Physica, 8D, 273-283, 1983.
- 4
- M. Bertsch, "A class of degenerate diffusion equations with a singular nonlinear term", Nonlinear Anal., 7, 117-127, 1983.
- 5
- R. Kersner, "Degenerate parabolic equations with general nonlinearities", Nonlinear Anal., 4, 1043-1062, 1980.
- 6
- T. Nakaki, K. Tomoeda, "A finite difference scheme for some nonlinear diffusion equations in an absorbing medium: support splitting phenomena", SIAM J. Numer. Anal., 40, 945-964, 2002.
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