To save this undefined to your undefined account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your undefined account.
Find out more about saving content to .
To save this article to your Kindle, first ensure email@example.com is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below.
Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
We consider the flow of a viscous incompressible fluid through a rigid
homogeneous porous medium. The permeability of the medium depends
on the pressure, so that the model is nonlinear. We propose a finite
element discretization of this problem and, in the case where the
dependence on the pressure is bounded from above and below, we prove
its convergence to the solution and propose an algorithm to solve
the discrete system. In the case where the dependence
on the pressure is exponential, we propose a splitting
scheme which involves solving two linear systems, but parts of the analysis of this method are still heuristic. Numerical tests are presented, which illustrate the introduced methods.
We consider the Pn model to approximate the time dependent transport equation in one dimension of space. In a diffusive regime, the solution of this system is solution of a diffusion equation.
We are looking for a numerical scheme having the diffusion limit property: in a diffusive regime, it has to give the solution of the limiting diffusion equation on a mesh at the diffusion scale.
The numerical scheme proposed is an extension of the Godunov type scheme proposed by Gosse to solve the P1 model without absorption term. It requires the computation of the solution of the steady state Pn equations. This is made by one Monte-Carlo simulation performed outside the time loop. Using formal expansions with respect to a small parameter representing the inverse of the number of mean free path in each cell, the resulting scheme is proved to have the diffusion limit. In order to avoid the CFL constraint on the time step, we give an implicit version of the scheme which preserves the positivity of the zeroth moment.
While alternans in a single cardiac cell appears through a simple
period-doubling bifurcation, in extended tissue the exact nature
of the bifurcation is unclear. In particular, the phase of
alternans can exhibit wave-like spatial dependence, either
stationary or travelling, which is known as discordant
alternans. We study these phenomena in simple cardiac models
through a modulation equation proposed by Echebarria-Karma. As
shown in our previous paper, the zero solution of their equation
may lose stability, as the pacing rate is increased, through
either a Hopf or steady-state bifurcation. Which bifurcation
occurs first depends on parameters in the equation, and for one
critical case both modes bifurcate together at a degenerate
(codimension 2) bifurcation. For parameters close to the
degenerate case, we investigate the competition between modes,
both numerically and analytically. We find that at sufficiently
rapid pacing (but assuming a 1:1 response is maintained), steady
patterns always emerge as the only stable solution. However, in
the parameter range where Hopf bifurcation occurs first, the
evolution from periodic solution (just after the bifurcation) to
the eventual standing wave solution occurs through an interesting
series of secondary bifurcations.
We present a model of the full thermo-mechanical evolution of a shape memory body undergoing a uniaxial tensile stress. The well-posedness of the related quasi-static thermo-inelastic problem is addressed by means of hysteresis operators techniques. As a by-product, details on a time-discretization of the problem are provided.
We introduce in this paper some elements for the mathematical and numerical analysis of algebraic turbulence models for
oceanic surface mixing layers. In these models the turbulent diffusions are parameterized by means of the gradient Richardson number, that measures the balance between stabilizing buoyancy forces and destabilizing shearing forces. We analyze the existence and linear exponential asymptotic stability of continuous and discrete equilibria states. We also analyze the well-posedness of a simplified model, by application of the linearization principle for non-linear parabolic equations. We finally present some numerical tests for realistic flows in tropical seas that reproduce the formation of mixing layers in time scales of the order of days, in agreement with the physics of the problem. We conclude that the typical mixing layers are transient effects due to the variability of equatorial winds. Also, that these states evolve to steady states in time scales of the order of years, under negative surface energy flux conditions.
When applied to the linear advection problem in dimension two, the
upwind finite volume method is a non consistent scheme in the finite
differences sense but a convergent scheme. According to our previous
paper [Bouche et al.,
SIAM J. Numer. Anal.43 (2005) 578–603], a sufficient condition in order to
complete the mathematical analysis of the finite volume scheme
consists in obtaining an estimation of order p, less or equal to
one, of a quantity that depends only on the mesh and on the advection
velocity and that we called geometric corrector. In [Bouche et al., Hermes Science publishing,
London, UK (2005) 225–236], we prove that, on the mesh given by
Peterson [SIAM J. Numer. Anal.28 (1991) 133–140] and for a subtle alignment of the
direction of transport parallel to the vertical boundary, the
infinite norm of the geometric corrector only behaves like h1/2
where h is a characteristic size of the mesh.
This paper focuses on the case of an oblique incidence i.e. a
transport direction that is not parallel to the boundary, still with
the Peterson mesh. Using various mathematical technics, we
explicitly compute an upper bound of the geometric corrector and we
provide a probabilistic interpretation in terms of Markov processes.
This bound is proved to behave like h, so that the order of
convergence is one. Then the reduction of the order of convergence
occurs only if the direction of advection is aligned with the
We analyze the accuracy and well-posedness of generalized impedance
boundary value problems in the framework of scattering problems
from unbounded highly absorbing media. We restrict ourselves in this first work
to the scalar problem (E-mode for electromagnetic scattering problems). Compared to earlier works, the unboundedness of the rough absorbing layer introduces severe difficulties
in the analysis for the generalized impedance boundary conditions, since
classical compactness arguments are no longer possible. Our new analysis
is based on the use of Rellich-type estimates and boundedness of L2
solution operators. We also discuss some numerical experiments
concerning these boundary conditions.
The paper is devoted to the computation of two-phase flows in a porous medium
when applying the two-fluid approach.
The basic formulation is presented first, together with the main properties
of the model. A few basic analytic solutions are then provided, some of them corresponding
to solutions of the one-dimensional Riemann problem.
Three distinct Finite-Volume schemes are then introduced. The first two schemes, which rely on the Rusanov scheme,
are shown to give wrong approximations in some cases
involving sharp porous profiles.
The third one, which is an extension of a scheme proposed by Kröner and Thanh [SIAM J. Numer. Anal.43 (2006) 796–824]
for the computation of single phase flows in varying cross section ducts,
provides fair results in all situations.
Properties of schemes and numerical results are presented.
Analytic tests enable to compute the L1 norm
of the error.