WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 23, 2024
Degenerate Parabolic Equation with Zero-flux Boundary Condition and its Approximations
Author:
Abstract: We study a degenerate parabolic-hyperbolic equation with zero-flux boundary condition. The aim of
this paper is to prove convergence of numerical approximate solutions towards the unique entropy solution. We
propose an implicit finite volume scheme on admissible mesh. We establish fundamental estimates and prove
that the approximate solution converge towards an entropy-process solution. Contrarily to the case of Dirichlet
condition, in zero-flux problem unnatural boundary regularity of the flux is required to establish that entropyprocess
solution is the unique entropy solution. In the study of well-posedness of the problem, tools of nonlinear
semigroup theory (stationary, mild and integral solutions) were used in order to overcome this difficulty. Indeed,
in some situations including the one-dimensional setting, solutions of the stationary problem enjoy additional
boundary regularity. Here, similar arguments are developed based on the new notion of integral-process solution
that we introduce for this purpose.
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Keywords: Hyperbolic-parabolic equation, Finite volume scheme, Zero-flux boundary condition, Convergence,
Boundary regularity, Entropy solution, Nonlinear semigroup theory, Mild solution, Integral-process solution
Pages: 682-705
DOI: 10.37394/23206.2024.23.71