43 research outputs found
Convergence of a Finite Volume Scheme for a Corrosion Model
In this paper, we study the numerical approximation of a system of partial
dif-ferential equations describing the corrosion of an iron based alloy in a
nuclear waste repository. In particular, we are interested in the convergence
of a numerical scheme consisting in an implicit Euler scheme in time and a
Scharfetter-Gummel finite volume scheme in space
Example of supersonic solutions to a steady state Euler-Poisson system
International audienceWe give an example of supersonic solutions to a one-dimensional steady state Euler-Poisson system arising in the modeling of plasmas and semiconductors. The existence of the supersonic solutions which correspond to large current density is proved by Schauder's fixed point theorem. We show also the uniqueness of solutions in the supersonic region
High order linearly implicit methods for semilinear evolution PDEs
This paper considers the numerical integration of semilinear evolution PDEs
using the high order linearly implicit methods developped in a previous paper
in the ODE setting. These methods use a collocation Runge--Kutta method as a
basis, and additional variables that are updated explicitly and make the
implicit part of the collocation Runge--Kutta method only linearly implicit. In
this paper, we introduce several notions of stability for the underlying
Runge--Kutta methods as well as for the explicit step on the additional
variables necessary to fit the context of evolution PDE. We prove a main
theorem about the high order of convergence of these linearly implicit methods
in this PDE setting, using the stability hypotheses introduced before. We use
nonlinear Schr\''odinger equations and heat equations as main examples but our
results extend beyond these two classes of evolution PDEs. We illustrate our
main result numerically in dimensions 1 and 2, and we compare the efficiency of
the linearly implicit methods with other methods from the litterature. We also
illustrate numerically the necessity of the stability conditions of our main
result
\^A-and \^I-stability of collocation Runge-Kutta methods
This paper deals with stability of classical Runge-Kutta collocation methods.
When such methods are embedded in linearly implicit methods as developed in
[12] and used in [13] for the time integration of nonlinear evolution PDEs, the
stability of these methods has to be adapted to this context. For this reason,
we develop in this paper several notions of stability, that we analyze. We
provide sufficient conditions that can be checked algorithmically to ensure
that these stability notions are fulfilled by a given Runge-Kutta collocation
method. We also introduce examples and counterexamples used in [13] to
highlight the necessity of these stability conditions in this context
The existence of solutions to a corrosion model
AbstractIn this work, we consider a corrosion model of iron based alloy in a nuclear waste repository. It consists of a PDE system, similar to the steady-state drift–diffusion system arising in semiconductor modelling. The main difference lies in the boundary conditions, since they are Robin boundary conditions and imply an additional coupling between the equations. Using a priori estimates for the solution and Schauder’s fixed point theorem, we show the existence of solutions to the corrosion model
A numerical study of vortex nucleation in 2D rotating Bose-Einstein condensates
This article introduces a new numerical method for the minimization under
constraints of a discrete energy modeling multicomponents rotating
Bose-Einstein condensates in the regime of strong confinement and with
rotation. Moreover, we consider both segregation and coexistence regimes
between the components. The method includes a discretization of a continuous
energy in space dimension 2 and a gradient algorithm with adaptive time step
and projection for the minimization. It is well known that, depending on the
regime, the minimizers may display different structures, sometimes with
vorticity (from singly quantized vortices, to vortex sheets and giant holes).
In order to study numerically the structures of the minimizers, we introduce in
this paper a numerical algorithm for the computation of the indices of the
vortices, as well as an algorithm for the computation of the indices of vortex
sheets. Several computations are carried out, to illustrate the efficiency of
the method, to cover different physical cases, to validate recent theoretical
results as well as to support conjectures. Moreover, we compare this method
with an alternative method from the literature
Global weak solutions to the compressible quantum navier-stokes equation and its semi-classical limit
International audienceThis paper is dedicated to the construction of global weak solutions to the quantum Navier-Stokes equation, for any initial value with bounded energy and entropy. The construction is uniform with respect to the Planck constant. This allows to perform the semi-classical limit to the associated compressible Navier-Stokes equation. One of the difficulty of the problem is to deal with the degenerate viscosity, together with the lack of integrability on the velocity. Our method is based on the construction of weak solutions that are renormalized in the velocity variable. The existence, and stability of these solutions do not need the Mellet-Vasseur inequality
On the existence of solutions for a drift-diffusion system arising in corrosion modelling
International audienceIn this paper, we consider a drift-diffusion system describing the corrosion of an iron based alloy in a nuclear waste repository. In comparison with the classical drift-diffusion system arising in the modeling of semiconductor devices, the originality of the corrosion model lies in the boundary conditions which are of Robin type and induce an additional coupling between the equations. We prove the existence of a weak solution by passing to the limit on a sequence of approximate solutions given by a semi-discretization in time