8,777 research outputs found
Role of non-ideality for the ion transport in porous media: derivation of the macroscopic equations using upscaling
This paper is devoted to the homogenization (or upscaling) of a system of
partial differential equations describing the non-ideal transport of a
N-component electrolyte in a dilute Newtonian solvent through a rigid porous
medium. Realistic non-ideal effects are taken into account by an approach based
on the mean spherical approximation (MSA) model which takes into account finite
size ions and screening effects. We first consider equilibrium solutions in the
absence of external forces. In such a case, the velocity and diffusive fluxes
vanish and the equilibrium electrostatic potential is the solution of a variant
of Poisson-Boltzmann equation coupled with algebraic equations. Contrary to the
ideal case, this nonlinear equation has no monotone structure. However, based
on invariant region estimates for Poisson-Boltzmann equation and for small
characteristic value of the solute packing fraction, we prove existence of at
least one solution. To our knowledge this existence result is new at this level
of generality. When the motion is governed by a small static electric field and
a small hydrodynamic force, we generalize O'Brien's argument to deduce a
linearized model. Our second main result is the rigorous homogenization of
these linearized equations and the proof that the effective tensor satisfies
Onsager properties, namely is symmetric positive definite. We eventually make
numerical comparisons with the ideal case. Our numerical results show that the
MSA model confirms qualitatively the conclusions obtained using the ideal model
but there are quantitative differences arising that can be important at high
charge or high concentrations.Comment: 46 page
On analysis error covariances in variational data assimilation
The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function (analysis). The equation for the analysis error is derived through the errors of the input data (background and observation errors). This equation is used to show that in a nonlinear case the analysis error covariance operator can be approximated by the inverse Hessian of an auxiliary data assimilation problem which involves the tangent linear model constraints. The inverse Hessian is constructed by the quasi-Newton BFGS algorithm when solving the auxiliary data assimilation problem. A fully nonlinear ensemble procedure is developed to verify the accuracy of the proposed algorithm. Numerical examples are presented
Computation of the unsteady facilitated transport of oxygen in hemoglobin
The transport of a reacting permeant diffusing through a thin membrane is extended to more realistic dissociation models. A new nonlinear analysis of the reaction-diffusion equations, using implicit finite-difference methods and direct block solvers, is used to study the limits of linearized and equilibrium theories. Computed curves of molecular oxygen permeating through hemoglobin solution are used to illustrate higher-order reaction models, the effect of concentration boundary layers at the membrane interfaces, and the transient buildup of oxygen flux
A study on iterative methods for solving Richards` equation
This work concerns linearization methods for efficiently solving the
Richards` equation,a degenerate elliptic-parabolic equation which models flow
in saturated/unsaturated porous media.The discretization of Richards` equation
is based on backward Euler in time and Galerkin finite el-ements in space. The
most valuable linearization schemes for Richards` equation, i.e. the
Newtonmethod, the Picard method, the Picard/Newton method and theLscheme are
presented and theirperformance is comparatively studied. The convergence, the
computational time and the conditionnumbers for the underlying linear systems
are recorded. The convergence of theLscheme is theo-retically proved and the
convergence of the other methods is discussed. A new scheme is
proposed,theLscheme/Newton method which is more robust and quadratically
convergent. The linearizationmethods are tested on illustrative numerical
examples
Oscillation-free method for semilinear diffusion equations under noisy initial conditions
Noise in initial conditions from measurement errors can create unwanted
oscillations which propagate in numerical solutions. We present a technique of
prohibiting such oscillation errors when solving initial-boundary-value
problems of semilinear diffusion equations. Symmetric Strang splitting is
applied to the equation for solving the linear diffusion and nonlinear
remainder separately. An oscillation-free scheme is developed for overcoming
any oscillatory behavior when numerically solving the linear diffusion portion.
To demonstrate the ills of stable oscillations, we compare our method using a
weighted implicit Euler scheme to the Crank-Nicolson method. The
oscillation-free feature and stability of our method are analyzed through a
local linearization. The accuracy of our oscillation-free method is proved and
its usefulness is further verified through solving a Fisher-type equation where
oscillation-free solutions are successfully produced in spite of random errors
in the initial conditions.Comment: 19 pages, 9 figure
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