[6] the LAD problem is proposed with a truncated Gaussian proﬁle as the initial condition.
In this paper we will solve the LAD problem with homogeneous boundary conditions and a sine proﬁle for the initial condition. This is exactly the same initial and boundary conditions that were imposed for the Burgers equation solved by Basdevant et al. [2] . We will be able to compare the physics associated with both prob- lems. The paper is organized as follows. Section 2 describes the LAD problem which is solved in closed form by the introduction of a change of variables. Section 3 details the analytical solution when the viscosity goes to zero. In this case the problem at hand is a simple wave equation perturbed by the presence of a very weak viscous term. Section 4 presents the Fourier solution when periodic conditions are applied. Section 5 is devoted to some con- siderations related to energy conservation. Section 6 treats the numerical method obtained by linear ﬁnite elements and a time integration using a Crank–Nicolson scheme for the viscous term and a second order Adams–Bashforth scheme for the **advection** term. Section 7 reports the results produced by both approaches and compares them. Finally the last section draws conclusions. 2. Linear **advection**–**diffusion** equation

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solution of the **advection**–**diffusion** transport equation using a change-of- variable and integral transform technique. Int J Heat Mass Trans 2009;52:3297–304 .
[11] Pérez Guérrero JS, Pimentel LCG, Skaggs TH, Van Genuchten MTh. Analytical solution of the one-dimensional **advection**–dispersion solute transport equation subject to time boundary conditions. Chem Eng J 2013;221:487–91 . [12] Sun P, Chen L, Xu J. Numerical studies of adaptive ﬁnite element methods for two dimensional convection–dominated problems. J Sci Comput 2010;43:24–43 .

Once again, we notice, see Fig 4, that the error is bigger inside the physical domain away from the boundary. This error thus only come from the discretization scheme and the PML scheme enables to efficiently treat **advection**-**diffusion** equation. From Table 2, we see that for Dirichlet and Neumann boundary conditions, the error is very strongly dominated by the truncation error.

Pascale Royer, Pascal Swider and Pauline Assemat
This work has been motivated by the study of solid tumor growth and more specifically by the examination of the most frequent pediatric primary bone tumors (osteosarcoma), which are characterised in their early stage by the for- mation of non-mineralised bone tissue, called osteoid [1]. As the tumor evolves in time, mineralisation of this growing tissue can take place. The purpose of this work thus is to investigate the **advection**-**diffusion** of soluble factor such as calcium and phosphate [2] in the context of tumor mineralisation, how this mineralisation of bone tissue may become a barrier to treatment in the context of drug transport [3] but also how proteins and growth factors can play a role in tumor growth [4]. For this purpose, the method of asymptotic homogenization is used so as to derive the macroscopic models for describing solute transport in poroelastic media, with an advective-diffusive regime in the fluid-saturated pores and **diffusion** in the solid phase, by starting from the description on the pore scale. The fluid/solid equations lead to Biot’s model of poroelasticity. Then, homogenization of the transport equations leads to three macroscopic models that relate to three orders of magnitude of the diffusivity ratio: a model in which the solid **diffusion** only influences the accumulation term; ii) a model with memory effects; iii) a model without solid **diffusion**. Initially expressed by means of orders of magnitude of the diffusivity ratio, the domains of validity of each of these three models can be expressed in terms of relative orders of magnitude of two characteristic times. The three models contain a solute-solid interaction term, due to the **advection** regime and they are coupled to the poroe- laticity model via the **advection** term.

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We investigate in this work the determination of the minimal time TM employing two distincts but complementary approaches. In a first one, we numerically estimate the cost of controllability, reformulated as the solution of a generalized eigenvalue problem for the underlying control operator, with respect to the parameter T and ε. This allows notably to exhibit the structure of initial data leading to large costs of control. At the practical level, this evaluation requires the non trivial and challenging approximation of null controls for the **advection**-**diffusion** equation. In the second approach, we perform an asymptotic analysis, with respect to the parameter ε, of the optimality system associated to the control of minimal L 2 -norm. The matched asymptotic expansion method is used to describe the multiple boundary layers.

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As ε goes to zero, the unique solution of the scalar **advection**-**diffusion** equation y ε
t − εy ε xx + M y ε x = 0, (x, t) ∈ (0, 1) × (0, T ) with Dirichlet boundary conditions exhibits a boundary layer of size O(ε) and an internal layer of size O( √ ε). If the time T is large enough, these thin layers where the solution y ε displays rapid variations intersect and interact with each other. Using the method of matched asymptotic expansions, we show how we can construct an explicit approximation e P ε of the solution y ε satisfying ky ε − e P ε kL ∞ (0,T ;L 2 (0,1)) = O(ε 3/2 ) and ky ε − e P ε kL 2 (0,T ;H 1 (0,1)) = O(ε), for all ε small enough.

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Abstract
In this work, we present a numerical analysis of a probabilistic approach to quantify the migration of a contaminant, under the presence of uncertainty on the permeability of the porous medium. More precisely, we consider the flow problem in a random porous medium coupled with the **advection**-**diffusion** equation and we are interested in the approximation of the mean spread and the mean dispersion of the solute. The conductivity field is represented by a Karhunen-Loève (K-L) decomposition of its logarithm. The flow model is solved using a mixed finite element method in the physical space. The **advection**- **diffusion** equation is computed thanks to a probabilistic particular method, where the concentration of the solute is the density function of a stochastic process. This process is solution of a stochastic differential equation (SDE), which is discretized using an Euler scheme. Then, the mean of the spread and of the dispersion are expressed as functions of the approximate stochastic process. A priori error estimates are established on the mean of the spread and of the dispersion. Numerical examples show the effectiveness of the approach.

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b Framatome ANP, Département Développement Codes & Méthodes – Tour AREVA, 1 Place Jean Millier, F-92400 Courbevoie, France. c Université d’Avignon et des Pays de Vaucluse, UMR 1114 EMMAH, 84018 Avignon Cedex, France.
Abstract
Partial differential equations (p.d.e) equipped with spatial derivatives of fractional order capture anomalous trans- port behaviors observed in diverse fields of Science. A number of numerical methods approximate their solutions in dimension one. Focusing our effort on such p.d.e. in higher dimension with Dirichlet boundary conditions, we present an approximation based on Lattice Boltzmann Method with Bhatnagar-Gross-Krook (BGK) or Multiple-Relaxation- Time (MRT) collision operators. First, an equilibrium distribution function is defined for simulating space-fractional **diffusion** equations in dimensions 2 and 3. Then, we check the accuracy of the solutions by comparing with i) ran- dom walks derived from stable Lévy motion, and ii) exact solutions. Because of its additional freedom degrees, the MRT collision operator provides accurate approximations to space-fractional **advection**-**diffusion** equations, even in the cases which the BGK fails to represent because of anisotropic **diffusion** tensor or of flow rate destabilizing the BGK LBM scheme.

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𝑥 .
Outline. In Section 2, we recall a few results regarding second-order differential operators of the **diffusion** type and their resolvent equations whose solutions are the Green functions: their factorized forms and their minimal pairs in particular. In Section 3, we provide our computation method which provide analytic closed form expression of the resolvent kernel (Green function). In Section 4, we treat in full detail the situation where the diffusivity and the advective term (or drift) are piecewise constant. We show how it reduces, through a simple change of variable, to the study of the drifted Skew Brownian motion. Using inverse Laplace transforms, we derive some explicit expressions for the transition density function (or the fundamental solution) for some particular situations. In Sections 5 and 6, we show that this general methodology is useful for solving **advection**-**diffusion** problems with particle tracking techniques. Section 5 details the case of one interface of discontinuity and Section 6 proposes a general algorithm to handle the case of multiple interfaces.

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When one computes the solution of this problem, it can only be solved numerically on a bounded domain. A good way to approximate the solution on the whole space may be given by the use of artificial boundary conditions (see [8, 9]).
In this paper, we will consider an **advection**-**diffusion** system in a strip Ω := R n−1 × (−L, 0) (L

ing to implement in the case of unsteady-state flow con- ditions, especially when lateral exchanges occur. The dif- fusive wave equation and the **advection**–**diffusion** transport equation have very similar mathematical expressions, but of course do not describe the same processes. The diffusive wave equation is derived from Saint-Venant continuity and momentum equations and can be applied to a wide range of phenomena in different fields as exposed by Singh (2002) for the kinematic wave, such as flood routing model but also for solute transport (Cimorelli et al., 2014). The **advection**– **diffusion** equation is derived from the mass conservation principle applied for matter dissolved in the water and tak- ing into account two basic processes of transport: **advection** and **diffusion** in which the Fick’s law leading to the diffu- sive term was applied. Under some hypotheses, the physi- cal equations of both the diffusive wave equation and the **advection**–**diffusion** equation can lead to the same similar mathematical expressions which can justify the use of the same resolution approaches. In the present study, following Singh (2002), the diffusive wave equation and the **advection**– **diffusion** equation are treated using the same mathematical approach: the Hayami (1951) analytical solution extended by Moussa (1996) to the case of uniformly lateral flow (and so- lute transport) using an inverse problem approach.

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MAR´ IA CRESPO, BENJAMIN IVORRA, ´ ANGEL MANUEL RAMOS Communicated by Jesus Ildefonso Diaz
Abstract. In this work, we present an asymptotic analysis of a coupled sys- tem of two **advection**-**diffusion**-reaction equations with Danckwerts boundary conditions, which models the interaction between a microbial population (e.g., bacteria), called biomass, and a diluted organic contaminant (e.g., nitrates), called substrate, in a continuous flow bioreactor. This system exhibits, under suitable conditions, two stable equilibrium states: one steady state in which the biomass becomes extinct and no reaction is produced, called washout, and another steady state, which corresponds to the partial elimination of the substrate. We use the linearization method to give sufficient conditions for the linear asymptotic stability of the two stable equilibrium configurations. Finally, we compare our asymptotic analysis with the usual asymptotic anal- ysis associated to the continuous bioreactor when it is modeled with ordinary differential equations.

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used to treat large populations. Instead some level of approximation has to be made to reduce the problem to a state equation in which the variable is the spatial density of individuals. Related works, concerning the transforma- tion of an individual-based or microscopic modelling into a population-based or macroscopic modelling, are Alt (1980) and Gr¨ unbaum (1999) in which the authors show that the solutions of an underlying differential-integral equation describing the movements of animals satisfy, under suitable assumptions, an **advection** **diffusion** equation. One can also be interested in Flierl et al. (1999) where the authors analyse the processes by which organisms form groups and discuss the transformation of IBM into continuum models. In the present study, an **advection**-**diffusion** equation is obtained as a truncated Kramers- Moyall cumulant expansion (Risken, 1996) of the spatial density function of individuals. The parameters of the IBM are used in the expressions of the ad- vection and **diffusion** terms. A consistent behavior is obtained concerning the dependence of these two terms on h and ∇h, and the balance between them. **Advection** and **diffusion** both are decreasing functions of the habitat index h . Moreover their dependence on ∇h implies that strong **advection** goes with weak **diffusion** leading to a directed movement of fish. On the contrary weak **advection** goes with strong **diffusion** corresponding to a searching behavior. This formalizes the heuristic approach of Faugeras and Maury (2005).

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1.3. Structure of the paper and main results. The main goal of the paper is to provide an existence result, with some extra estimates, for the Fokker-Planck equation ( 1.5 ) via time discretization, using the so-called splitting method (the two main ingredients of the equation, i.e. the **advection** with **diffusion** on one hand, and the density constraint on the other hand, are treated one after the other). In Section 2 we will collect some preliminary results, including what we need from optimal transport and from the previous works about density-constrained crowd motion, in particular on the projection operator onto the set K. In Section 3 we will provide the existence result we aim at, by a splitting scheme and some entropy bounds; the solution will be a curve of measures in AC 2 ([0, T ]; W2 (Ω)) (absolutely continuous curves with square-integrable speed). In Section 4 we will make use of BV estimates to justify that the solution we just built is also Lip([0, T ]; W 1 (Ω)) and satisfies a global BV bound kρ t k BV ≤ C (provided ρ 0 ∈ BV ): this requires to combine BV estimates on the Fokker-Planck equation (which are available depending on the regularity of the vector field u) with BV estimates on the projection operator on K (which have been recently proven in [ 14 ]). Section 5 presents a short review of alternative approaches, all discretized in time, but based either on gradient-flow techniques (the JKO scheme, see [ 18 ]) or on different splitting methods. Finally, in the Appendix A we detail the BV estimates on the Fokker-Planck equation (without any density constraint) that we could find; this seems to be a delicate matter, interesting in itself, and we are not aware of the sharp assumptions on the vector field u to guarantee the BV estimate that we need.

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Unité de recherche INRIA Rocquencourt Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex France Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois -[r]

Unité de recherche Inria Lorraine, Technopôle de Nancy-Brabois, Campus scientifique, 615 rue du Jardin Botanique, BP 101, 54600 Villers Lès Nancy Unité de recherche Inria Rennes, Irisa, [r]

not fall in the framework of the weak convergence result stated above as it depends on ε ! Nevertheless, the time T M and more generally the behavior of the control of minimal L 2 -norm (which appears in (3))
remains unclear for ε small: there is a kind of balance between the term −εy ε
xx which favor the **diffusion**

issues. The ﬁrst issue is how enriching the approach with bubble functions aﬀects the accuracy. Of course, as all our approaches are variational approaches, enriching the variational approximation space using bubble functions can only improve the accuracy, at the price of increasing the number of degrees of freedom. We however notice that, when a gain in accuracy is observed, that gain is much higher than that obtained by, say, reducing by a factor two the size of the coarse mesh. We therefore then safely conclude about the added value of bubble functions. Other issues that are speciﬁcally examined below are the inﬂuence of the Péclet number (measuring the relative amplitude of the **advection** with respect to the diﬀusion) and that of the small scale ε deﬁning both the size of the perforations and their typical distance. Many of these issues are examined upon considering a range of mesh sizes H for the coarse mesh. This range is typically chosen as H varying from ε{10 to 10 ε. One must bear in mind that capturing all the details of the oscillatory solutions u ε using a standard FEM approach would require choosing a mesh size in

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Unité de recherche INRIA Sophia Antipolis 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex France Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - [r]

Here we are interested in the case of a lognormal permeability eld, which is a widely used model. Moreover we consider the case, physically pertinent, where the correlation length is small and the uncertainty important. Therefore methods based on the approximation of the coecients in a nite dimensional stochastic space, such as stochastic galerkin methods and stochastic collocation method would be highly expensive, and hence do not seem to be suitable to deal with such cases. Neither seem perturbation methods, since we suppose the uncertainty to be important. As regards the **advection**-diusion equation, we focus on the **advection**-dominated model. Therefore we choose not to consider an Eulerian method, in order to avoid numerical diusion. The below described method has therefore being proposed and implemented by A.Beaudoin, J.R. de Dreuzy and J.Erhel to compute the mean dispersion in 2D, their numerical results can be found e.g. in [6]. A Monte-Carlo method is used to deal with the uncertainty. The solution of the steady ow equation is computed by using nite elements. The solution of the **advection**-diusion equation is approximated using a probabilistic method. We consider the stochastic dierential equation associated to this Fokker Planck equation and its solution is approximated with an Euler scheme. A Monte-Carlo method provides nally an approximation to the solution of the Fokker Planck equation. All these steps together lead to an approximation of the mean spread. The mean dispersion is then approximated by the numerical derivative of the computed mean spread. The aim of this paper is to make the numerical analysis of the above described method. More precisely we furnish a priori error estimates for the approximations of the spread and of the dispersion. We focus on the spread and the dispersion, because of their physical interest, but the result given here are more general. A specity of this work is to address the coupling of the ow equation with the **advection**-diusion equation, whereas most of the existing numerical analysis of methods for uncertainty quantication are limited to the ow equation, see e.g [1], [2]. The main novelty of this work is the use of numerical analysis tools from two dierents areas: nite elements method and weak error analysis for SDEs. Moreover, since we estimate the time derivative of the spread, we have to generalize the weak error to estimate the error for time derivatives of averages.

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