216 research outputs found
Numerical analysis of a robust free energy diminishing Finite Volume scheme for parabolic equations with gradient structure
We present a numerical method for approximating the solutions of degenerate
parabolic equations with a formal gradient flow structure. The numerical method
we propose preserves at the discrete level the formal gradient flow structure,
allowing the use of some nonlinear test functions in the analysis. The
existence of a solution to and the convergence of the scheme are proved under
very general assumptions on the continuous problem (nonlinearities, anisotropy,
heterogeneity) and on the mesh. Moreover, we provide numerical evidences of the
efficiency and of the robustness of our approach
Convergence of finite volume scheme for degenerate parabolic problem with zero flux boundary condition
This note is devoted to the study of the finite volume methods used in the discretization of degenerate parabolic-hyperbolic equation with zero-flux boundary condition. The notion of an entropy-process solution, successfully used for the Dirichlet problem, is insufficient to obtain a uniqueness and convergence result because of a lack of regularity of solutions on the boundary. We infer the uniqueness of an entropy-process solution using the tool of the nonlinear semigroup theory by passing to the new abstract notion of integral-process solution. Then, we prove that numerical solution converges to the unique entropy solution as the mesh size tends to 0
Well-posedness results for triply nonlinear degenerate parabolic equations
We study the well-posedness of triply nonlinear degenerate
elliptic-parabolic-hyperbolic problem in a bounded domain with
homogeneous Dirichlet boundary conditions. The nonlinearities and
are supposed to be continuous non-decreasing, and the nonlinearity
falls within the Leray-Lions framework. Some restrictions
are imposed on the dependence of on
and also on the set where degenerates. A model case is
with which is strictly increasing except on a locally finite number of
segments, and which is of the Leray-Lions kind. We are
interested in existence, uniqueness and stability of entropy solutions. If
, we obtain a general continuous dependence result on data
and nonlinearities . Similar result
is shown for the degenerate elliptic problem which corresponds to the case of
and general non-decreasing surjective . Existence, uniqueness
and continuous dependence on data are shown when and
is continuous
Explicit formulation for the Dirichlet problem for parabolic-hyperbolic conservation laws
We revisit the Cauchy-Dirichlet problem for degenerate parabolic scalar conservation laws. We suggest a new notion of strong entropy solution. It gives a straightforward explicit characterization of the boundary values of the solution and of the flux, and leads to a concise and natural uniqueness proof, compared to the one of the fundamental work [J. Carrillo, Arch. Ration. Mech. Anal., 1999]. Moreover, general dissipative boundary conditions can be studied in the same framework. The definition makes sense under the specific weak trace-regularity assumption. Despite the lack of evidence that generic solutions are trace-regular (especially in space dimension larger than one), the strong entropy formulation may be useful for modeling and numerical purposes
Solutions processus intégrales des équations d'évolution abstraites et application à l'approximation numérique d'un problème parabolique dégénéré. [ Integral-process solutions of abstract evolution equations and application to numerical approximation of a degenerate parabolic boundary-value problem. ]
bilingual (French with abriged English version)(French) Nous introduisons la notion de solution processus intégrale pour un problème d'évolution , gouverné par un opérateur -accrétif dans un espace de Banach . Nous prouvons qu'une telle solution coïncide avec l'unique solution intégrale du problème. Ce résultat technique appliqué avec dans une approche de compacité faible du type mesures de Young permet de montrer la convergence d'un schéma volumes finis que nous avons construit pour l'équation parabolique-hyperbolique munie de la condition de flux zéro sur le bord. (English) We introduce a notion of integral-process solution for evolution problem , governed by an -accrétive operator in a Banach space . We prove that such solution coincides with the unique integral solution of the problem. Applying this technical result with within a weak compactness approach of Young measures' type, we prove convergence of a suitably defined finite volume scheme to the unique entropy solution of the parabolic-hyperbolic equation with the zero-flux boundary condition
Degenerate parabolic equation with zero flux boundary condition and its approximations
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 conditions, in
zero-flux problem unnatural boundary regularity of the flux is required to
establish that entropy-process 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 [Andreianov, Gazibo,
ZAMP, 2013] 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.Comment: 41 page
Adaptive multiresolution schemes with local time stepping for two-dimensional degenerate reaction-diffusion systems
We present a fully adaptive multiresolution scheme for spatially
two-dimensional, possibly degenerate reaction-diffusion systems, focusing on
combustion models and models of pattern formation and chemotaxis in
mathematical biology. Solutions of these equations in these applications
exhibit steep gradients, and in the degenerate case, sharp fronts and
discontinuities. The multiresolution scheme is based on finite volume
discretizations with explicit time stepping. The multiresolution representation
of the solution is stored in a graded tree. By a thresholding procedure, namely
the elimination of leaves that are smaller than a threshold value, substantial
data compression and CPU time reduction is attained. The threshold value is
chosen optimally, in the sense that the total error of the adaptive scheme is
of the same slope as that of the reference finite volume scheme. Since chemical
reactions involve a large range of temporal scales, but are spatially well
localized (especially in the combustion model), a locally varying adaptive time
stepping strategy is applied. It turns out that local time stepping accelerates
the adaptive multiresolution method by a factor of two, while the error remains
controlled.Comment: 27 pages, 14 figure
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