1,404 research outputs found
The discontinuous Galerkin method for fractional degenerate convection-diffusion equations
We propose and study discontinuous Galerkin methods for strongly degenerate
convection-diffusion equations perturbed by a fractional diffusion (L\'evy)
operator. We prove various stability estimates along with convergence results
toward properly defined (entropy) solutions of linear and nonlinear equations.
Finally, the qualitative behavior of solutions of such equations are
illustrated through numerical experiments
Numerical analysis of a nonlinear free-energy diminishing Discrete Duality Finite Volume scheme for convection diffusion equations
We propose a nonlinear Discrete Duality Finite Volume scheme to approximate
the solutions of drift diffusion equations. The scheme is built to preserve at
the discrete level even on severely distorted meshes the energy / energy
dissipation relation. This relation is of paramount importance to capture the
long-time behavior of the problem in an accurate way. To enforce it, the linear
convection diffusion equation is rewritten in a nonlinear form before being
discretized. We establish the existence of positive solutions to the scheme.
Based on compactness arguments, the convergence of the approximate solution
towards a weak solution is established. Finally, we provide numerical evidences
of the good behavior of the scheme when the discretization parameters tend to 0
and when time goes to infinity
Residual equilibrium schemes for time dependent partial differential equations
Many applications involve partial differential equations which admits
nontrivial steady state solutions. The design of schemes which are able to
describe correctly these equilibrium states may be challenging for numerical
methods, in particular for high order ones. In this paper, inspired by
micro-macro decomposition methods for kinetic equations, we present a class of
schemes which are capable to preserve the steady state solution and achieve
high order accuracy for a class of time dependent partial differential
equations including nonlinear diffusion equations and kinetic equations.
Extension to systems of conservation laws with source terms are also discussed.Comment: 23 pages, 12 figure
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
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
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