Different choices of scaling in homogenization of diffusion and interfacial exchange in a porous medium

Several choices of scaling are investigated for a coupled system of parabolic partial differential equations in a two‐phase medium at the microscopic scale. This system may be regarded as modelling a reaction–diffusion problem, the Stokes problem of single‐phase flow of a slightly compressible fluid or as a heat conduction problem (with or without interfacial resistance), for example. It is shown that, starting with the same problem on the microscopic scale, different choices of scaling of the diffusion coefficients (resp. permeability or conductivity) and the interfacial‐exchange coefficient lead to different types of macroscopic systems of equations. The characterization of the limit problems in terms of the scaling parameters constitutes a modelling tool because it allows to determine the right type of limit problem. New macroscopic models, not previously dealt with, arise and, for some scalings, classical macroscopic models are recovered. Using the method of two‐scale convergence, a unified approach yielding rigorous proofs is given covering a very broad class of different scalings. Copyright © 2008 John Wiley & Sons, Ltd.     

Some asymptotic limits of reaction–diffusion systems appearing in reversible chemistry

This paper concerns reaction–diffusion systems consisting of three or four equations, which come out of reversible chemistry. We introduce different scalings for those systems, which make sense in various situations (species with very different concentrations or very different diffusion rates, chemical reactions with very different rates, etc.). We show how recently introduced mathematical tools allow to prove that the formal asymptotics associated to those scalings indeed hold at the rigorous level.     

Asymptotic Analysis of the Current-Voltage Curve of a pnpn Semiconductor Device

We consider the one-dimensional steady-state semiconductor deviceequations modelling a pnpn device. There are two relevant scalingsof the equations corresponding to small and large applied voltages.In both scalings, the semiconductor equations can be consideredas singularly perturbed. It turns out that the small-voltagescaling breaks down for current values between two saturationcurrents. In that interval, the large-voltage scaling has tobe employed. For both scalings, we derive the first-order termsof an asymptotic expansion and show that the reduced problemhas a solution. An example verifies that the current-voltagecurves obtained have the expected qualitative structure.     

Catalytic Discrete State Branching Models and Related Limit Theorems

Catalytic discrete state branching processes with immigration are defined as strong solutions of stochastic integral equations. We provide main limit theorems of those processes using different scalings. The class of limit processes of the theorems includes essentially all continuous state catalytic branching processes and spectrally positive regular affine processes.      

A Two‐Parameter Stabilized Finite Element Method for Incompressible Flows

Based on two‐grid discretizations, a two‐parameter stabilized finite element method for the steady incompressible Navier–Stokes equations at high Reynolds numbers is presented and studied. In this method, a stabilized Navier–Stokes problem is first solved on a coarse grid, and then a correction is calculated on a fine grid by solving a stabilized linear problem. The stabilization term for the nonlinear Navier–Stokes equations on the coarse grid is based on an elliptic projection, which projects higher‐order finite element interpolants of the velocity into a lower‐order finite element interpolation space. For the linear problem on the fine grid, either the same stabilization approach (with a different stabilization parameter) as that for the coarse grid problem or a completely different stabilization approach could be employed. Error bounds for the discrete solutions are estimated. Algorithmic parameter scalings of the method are also derived. The theoretical results show that, with suitable scalings of the algorithmic parameters, this method can yield an optimal convergence rate. Numerical results are provided to verify the theoretical predictions and demonstrate the effectiveness of the proposed method. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 425–444, 2017     

The Distance Perspective of Generalized Biadditive Models: Scalings and Transformations

Recently two articles studied scalings in biplot models, and concluded that these have little impact on the interpretation. In this article again scalings are studied for generalized biadditive models and correspondence analysis, that is, special cases of the general biplot family, but from a different perspective. The generalized biadditive models, but also correspondence analysis, are often used for Gaussian ordination. In Gaussian ordination one takes a distance perspective for the interpretation of the relationship between a row and a column category. It is shown that scalings—but also nonsingular transformations—have a major impact on this interpretation. So, depending on the perspective one takes, the inner product or distance perspective, scalings and transformations do have (distance) or do not have (inner-product) impact on the interpretation. If one is willing to go along with the assumption of the author that diagrams are in practice often interpreted by a distance rule, the findings in this article influence all biplot models.     

Reaction-diffusion models from the Fokker-Planck formulation ofchemical processes

A Fokker-Planck-type model is proposed to describe the kineticsof certain chemical reactions. In particular, the competitionbetween transport and reaction processes is analysed. The studyis carried out considering various scalings of interaction,measured by the exponent of a small parameter related to themean free path. In the most significant case of competitionbetween both effects, the lowest-order density in the asymptoticexpansion obeys a reaction-diffusion equation. Such an equationwas earlier considered as the starting point in the study ofthese processes by other authors (e.g. by Schlgl). For otherinteraction scalings, the prevalence of chemical processes impliesthat the lowest-order density is determined by the (algebraic)equations of chemical equilibrium. In contrast, when transportprevails, the reaction terms affect only higher-orderdensities.     

Finite element analysis of the vibration problem of a plate coupled with a fluid

Summary. We consider the approximation of the vibration modes of an elastic plate in contact with a compressible fluid. The plate is modelled by Reissner-Mindlin equations while the fluid is described in terms of displacement variables. This formulation leads to a symmetric eigenvalue problem. Reissner-Mindlin equations are discretized by a mixed method, the equations for the fluid with Raviart-Thomas elements and a non conforming coupling is used on the interface. In order to prove that the method is locking free we consider a family of problems, one for each thickness , and introduce appropriate scalings for the physical parameters so that these problems attain a limit when . We prove that spurious eigenvalues do not arise with this discretization and we obtain optimal order error estimates for the eigenvalues and eigenvectors valid uniformly on the thickness parameter t. Finally we present numerical results confirming the good performance of the method. Received February 4, 1998 / Revised version received May 26, 1999 / Published online June 21, 2000     

An explicitly solvable kinetic model for vehicular traffic and associated macroscopic equations

In the present paper, a kinetic model for vehicular traffic is presented and investigated in detail. For this model, the stationary distributions can be determined explicitly. A derivation of associated macroscopic traffic flow equations from the kinetic equation is given. The coefficients appearing in these equations are identified from the solutions of the underlying stationary kinetic equation and are given explicitly. Moreover, numerical experiments and comparisons between different macroscopic models are presented.     

Analysis of a micro–macro acceleration method with minimum relative entropy moment matching

We analyse convergence of a micro–macro acceleration method for the simulation of stochastic differential equations with time-scale separation. The method alternates short bursts of path simulations with the extrapolation of macroscopic state variables forward in time. After extrapolation, a new microscopic state is constructed, consistent with the extrapolated macroscopic state, that minimises the perturbation caused by the extrapolation in a relative entropy sense. We study local errors and numerical stability of the method to prove its convergence to the full microscopic dynamics when the extrapolation time step tends to zero and the number of macroscopic state variables tends to infinity.     

On the Interplay among Entropy, Variable Metrics and Potential Functions in Interior-Point Algorithms

We are motivated by the problem of constructing aprimal-dual barrier function whose Hessian induces the (theoreticallyand practically) popular symmetric primal and dual scalings forlinear programming problems. Although this goal is impossible toattain, we show that the primal-dual entropy function may provide asatisfactory alternative. We study primal-dual interior-pointalgorithms whose search directions are obtained from a potentialfunction based on this primal-dual entropy barrier. We providepolynomial iteration bounds for these interior-point algorithms. Thenwe illustrate the connections between the barrier function and areparametrization of the central path equations. Finally, we considerthe possible effects of more general reparametrizations oninfeasible-interior-point algorithms.     

Numerical simulation of biofilm growth in porous media

Biofilms are very important in controlling pollution in aquifers. The bacteria may either consume the contaminant or form biobarriers to limit its spread. In this paper we review the mathematical modeling of biofilm growth at the microscopic and macroscopic scales, together with a scale-up technique. At the pore-scale, we solve the Navier-Stokes equations for the flow, the advection-diffusion equation for the transport, together with equations for the biofilm growth. These results are scaled up using network model techniques, in order to have relations between the amount and distribution of the biomass, and macroscopic properties such as permeability and porosity. A macroscopic model is also presented. We give some results.     

The macroscopic system of Einstein-Maxwell equations for a system of interacting particles

We derive the macroscopic Einstein—Maxwell equations up to the second-order terms, in the interaction for systems with dominating electromagnetic interactions between particles (e.g., radiation-dominated cosmological plasma in the expanding Universe before the recombination moment). The ensemble averaging of the microscopic Einstein and Maxwell equations and of the Liouville equations for the random functions of each type of particle leads to a closed system of equations consisting of the macroscopic Einstein and Maxwell equations and the kinetic equations for one-particle distribution functions for each type of particle. The macroscopic Einstein equations for a system of electromagnetically and gravitationally interacting particles differ from the classical Einstein equations in having additional terms in the lefthand side due to the interaction. These terms are given by a symmetric rank-two traceless tensor with zero divergence. Explicitly, these terms are represented as momentum-space integrals of the expressions containing one-particle distribution functions for each type of particle and have much in common with similar terms in the left-hand side of the macroscopic Einstein equations previously obtained for a system of self-gravitating particles. The macroscopic Maxwell equations for a system of electromagnetically and gravitationally interacting particles also differ from the classical Maxwell equations in having additional terms in the left-hand side due to simultaneous effects described by general relativity and the interaction effects. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 125, No. 1, pp. 107–131, October, 2000.     

Solving Multi‐Scale problems with heterogeneous finite difference methods

Multi‐scale differential equations are problems in which the variables can have different length scales. The direct numerical solution of differential equations with multiple scales is often difficult due to the work for resolving the smallest scale. We present here a strategy which allows the use of finite difference methods for the numerical solution of parabolic multi‐scale problems, based on a coupling of macroscopic and microscopic models for the original equation.     

An improved macroscopic model of traffic flow: Derivation and links with the Lighthill-Whitham model

We first recall a macroscopic model with two equations, that we have introduced in [1], and which completely resolves the severe inconsistencies of the class of Payne-Whitham models. In this short paper, we describe the effects of adding a relaxation term in the anticipation equation, and the main steps and mathematical difficulties to show rigorously the convergence to the Lighthill-Whitham model when the relaxation time tends to 0.     

The Dynamics of a Conserved Phase Field System: Stefan-like, Hele-Shaw, and Cahn-Hilliard Models as Asymptotic Limits

The dynamics of a conserved phase field system for a free boundary,that is, are studied asymptotically.Many of the major macroscopic free-boundary problems arise aslimits in various scalings. Temperature, curvature, surfacetension, and velocity relations are derived, and compared withanalogous results for systems using a nonconserved order parameter.A single fourth-order equation which has been studied in spinodaldecomposition (Cahn-Hilliard) is obtained from this system ofequations by setting the latent heat to zero.     

On transport equations in slow flows of a polyatomic gas in an external magnetic field

The transport equations in slow flows of a polyatomic gas in an external magnetic field varying with time are obtained by quantum kinetic theory methods. It is shown that local macroscopic variables, changing slowly with time, obey a system of linked equations. In the Burnett approximation, the Wigner operator and stress tensor are obtained. Transport coefficients depending on the magnetic field frequency are determined. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 3, pp. 459–469, March, 1997.     

Macroscopic regularity for the Boltzmann equation

The regularity of solutions to the Boltzmann equation is a fundamental problem in the kinetic theory. In this paper, the case with angular cut-off is investigated. It is shown that the macroscopic parts of solutions to the Boltzmann equation, i.e., the density, momentum and total energy are continuous functions of(x, t) in the region R3×(0, +∞). More precisely, these macroscopic quantities immediately become continuous in any positive time even though they are initially discontinuous and the discontinuities of solutions propagate only in the microscopic level. It should be noted that such kind of phenomenon can not happen for the compressible Navier-Stokes equations in which the initial discontinuities of the density never vanish in any finite time, see [22]. This hints that the Boltzmann equation has better regularity effect in the macroscopic level than compressible Navier-Stokes equations.