# American Institute of Mathematical Sciences

ISSN:
1556-1801

eISSN:
1556-181X

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## Networks & Heterogeneous Media

September 2010 , Volume 5 , Issue 3

A special issue
New Trends in Model Coupling, Theory, Numerics and Applications

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2010, 5(3): i-ii doi: 10.3934/nhm.2010.5.3i +[Abstract](1955) +[PDF](46.6KB)
Abstract:
Special Issue from the workshop New Trends in Model Coupling, Theory, Nu- merics and Applications (NTMC’09), Paris, September 2 − 4 2009.
This special issue comprises selected papers from the workshop New Trends in Model Coupling, Theory, Numerics and Applications (NTMC'09) which took place in Paris, September 2 - 4, 2009. The research of optimal technological solutions in a large amount of industrial systems requires to perform numerical simulations of complex phenomena which are often characterized by the coupling of models related to various space and/or time scales. Thus, the so-called multiscale modelling has been a thriving scientific activity which connects applied mathematics and other disciplines such as physics, chemistry, biology or even social sciences. To illustrate the variety of fields concerned by the natural occurrence of model coupling we may quote:
• meteorology where it is required to take into account several turbulence scales or the interaction between oceans and atmosphere, but also regional models in a global description,
• solid mechanics where a thorough understanding of complex phenomena such as propagation of cracks needs to couple various models from the atomistic level to the macroscopic level;
• plasma physics for fusion energy for instance where dense plasmas and collisionless plasma coexist;
• multiphase fluid dynamics when several types of flow corresponding to several types of models are present simultaneously in complex circuits;
• social behaviour analysis with interaction between individual actions and collective behaviour.

2010, 5(3): 385-404 doi: 10.3934/nhm.2010.5.385 +[Abstract](1757) +[PDF](688.0KB)
Abstract:
We present in this paper several results concerning a simple model of interaction between an inviscid fluid, modeled by the Burgers equation, and a particle, assumed to be point-wise. It is composed by a first-order partial differential equation which involves a singular source term and by an ordinary differential equation. The coupling is ensured through a drag force that can be linear or quadratic. Though this model can be considered as a simple one, its mathematical analysis is involved. We put forward a notion of entropy solution to our model, define a Riemann solver and make first steps towards well-posedness results. The main goal is to construct easy-to-implement and yet reliable numerical approximation methods; we design several finite volume schemes, which are analyzed and tested.
2010, 5(3): 405-422 doi: 10.3934/nhm.2010.5.405 +[Abstract](1804) +[PDF](231.8KB)
Abstract:
We propose here a general framework to address the question of trace operators on a dyadic tree. This work is motivated by the modeling of the human bronchial tree which, thanks to its regularity, can be extrapolated in a natural way to an infinite resistive tree. The space of pressure fields at bifurcation nodes of this infinite tree can be endowed with a Sobolev space structure, with a semi-norm which measures the instantaneous rate of dissipated energy. We aim at describing the behaviour of finite energy pressure fields near the end. The core of the present approach is an identification of the set of ends with the ring $\ZZ$2 of 2-adic integers. Sobolev spaces over $\ZZ$2 can be defined in a very natural way by means of Fourier transform, which allows us to establish precised trace theorems which are formally quite similar to those in standard Sobolev spaces, with a Sobolev regularity which depends on the growth rate of resistances, i.e. on geometrical properties of the tree. Furthermore, we exhibit an explicit expression of the "ventilation operator'', which maps pressure fields at the end of the tree onto fluxes, in the form of a convolution by a Riesz kernel based on the 2-adic distance.
2010, 5(3): 423-460 doi: 10.3934/nhm.2010.5.423 +[Abstract](2062) +[PDF](544.0KB)
Abstract:
The purpose of this article is to present a unified view of some multiscale models that have appeared in the past decades in computational materials science. Although very different in nature at first sight, since they are employed to simulate complex fluids on the one hand and crystalline solids on the other hand, the models presented actually share a lot of similarities, many of those being in fact also present in most multiscale strategies. The mathematical and numerical difficulties that these models generate, the way in which they are utilized (in particular as numerical strategies coupling different models in different regions of the computational domain), the computational load they imply, are all very similar in nature. In particular, a common feature of these models is that they require knowledge and techniques from different areas in Mathematics: theory of partial differential equations, of ordinary differential equations, of stochastic differential equations, and all the related numerical techniques appropriate for the simulation of these equations. We believe this is a general trend of modern computational modelling.
2010, 5(3): 461-485 doi: 10.3934/nhm.2010.5.461 +[Abstract](1907) +[PDF](950.7KB)
Abstract:
We study a system of conservation laws that describes multi-species kinematic flows with an emphasis on models of multiclass traffic flow and of the creaming of oil-in-water dispersions. The flux can have a spatial discontinuity which models abrupt changes of road surface conditions or of the cross-sectional area in a settling vessel. For this system, an entropy inequality is proposed that singles out a relevant solution at the interface. It is shown that "piecewise smooth" limit solutions generated by the semi-discrete version of a numerical scheme the authors recently proposed [R. Bürger, A. García, K.H. Karlsen and J.D. Towers, J. Engrg. Math. 60:387-425, 2008] satisfy this entropy inequality. We present an improvement to this scheme by means of a special interface flux that is activated only at a few grid points where the discontinuity is located. While an entropy inequality is established for the semi-discrete versions of the scheme only, numerical experiments support that the fully discrete scheme are equally entropy-admissible.
2010, 5(3): 487-505 doi: 10.3934/nhm.2010.5.487 +[Abstract](2158) +[PDF](521.5KB)
Abstract:
We introduce nonoverlapping domain decomposition algorithms of Schwarz waveform relaxation type for the semilinear reaction-diffusion equation. We define linear Robin and second order (or Ventcell) transmission conditions between the subdomains, which we prove to lead to a well defined and converging algorithm. We also propose nonlinear transmission conditions. Both types are based on best approximation problems for the linear equation and provide efficient algorithms, as the numerical results that we present here show.
2010, 5(3): 507-524 doi: 10.3934/nhm.2010.5.507 +[Abstract](1696) +[PDF](233.5KB)
Abstract:
This paper is devoted to the study of the one dimensional interfacial coupling of two PDE systems at a given fixed interface, say $x=0$. Each system is posed on a half-space, namely $x<0$ and $x>0$. As an interfacial model, a coupling condition whose objective is to enforce the continuity (in a weak sense) of a prescribed variable is generally imposed at $x=0$.
We first focus on the coupling of two scalar conservation laws and state an existence result for the coupled Riemann problem. Numerical experiments are also proposed. We then consider, both from a theoretical and a numerical point of view, the coupling of two-phase flow models namely a drift-flux model and a two-fluid model. In particular, the link between both models will be addressed using asymptotic expansions.
2010, 5(3): 525-544 doi: 10.3934/nhm.2010.5.525 +[Abstract](2269) +[PDF](3304.7KB)
Abstract:
This work deals with the modelling of traffic flows in complex networks, spanning two-dimensional regions whose size ( macroscale ) is much greater than the characteristic size of the network arcs ( microscale). A typical example is the modelling of traffic flow in large urbanized areas with diameter of hundreds of kilometers, where standard models of traffic flows on networks resolving all the streets are computationally too expensive. Starting from a stochastic lattice gas model with simple constitutive laws, we derive a distributed two-dimensional model of traffic flow, in the form of a non-linear diffusion-advection equation for the particle density. The equation is formally equivalent to a (non-linear) Darcy's filtration law. In particular, it contains two parameters that can be seen as the porosity and the permeability tensor of the network. We provide suitable algorithms to extract these parameters starting from the geometry of the network and a given microscale model of traffic flow (for instance based on cellular automata). Finally, we compare the fully microscopic simulation with the finite element solution of our upscaled model in realistic cases, showing that our model is able to capture the large-scale feature of the flow.
2010, 5(3): 545-563 doi: 10.3934/nhm.2010.5.545 +[Abstract](1506) +[PDF](5497.3KB)
Abstract:
We present different ways, coming from Finite Volume or Mixed Finite Element frameworks, to discretize convection terms in Hybrid Finite Volume, Mimetic Finite Difference and Mixed Finite Volume methods for elliptic equations. We compare them through several numerical tests, deducing some generic principles, depending on the situation, on the choice of an apropriate method and its parameters. We also present an adaptation to the Navier-Stokes equations, with a numerical tests in the case of the lid-driven cavity.
2010, 5(3): 565-581 doi: 10.3934/nhm.2010.5.565 +[Abstract](2468) +[PDF](275.8KB)
Abstract:
This paper deals with various applications of conservation laws on networks. In particular we consider the car traffic, described by the Lighthill-Whitham-Richards model and by the Aw-Rascle-Zhang model, the telecommunication case, by using the model introduced by D'Apice-Manzo-Piccoli and, finally, the case of a gas pipeline, modeled by the classical $p$-system. For each of these models we present a review of some results about Riemann and Cauchy problems in the case of a network, formed by a single vertex with $n$ incoming and $m$ outgoing arcs.
2010, 5(3): 583-602 doi: 10.3934/nhm.2010.5.583 +[Abstract](2365) +[PDF](649.4KB)
Abstract:
With the depletion of oil reserves and increase in oil price, enhanced oil recovery methods such as polymer flooding to increase oil production from waterflooded fields are becoming more attractive. Effective design of these processes is challenging because the polymer chemistry has a strong effect on reaction and fluid rheology, which in turn has a strong effect on fluid transport. Polymer flow characteristics modeled in the UT-Austin IPARS (Integrated Parallel Accurate Reservoir Simulator) are adsorption on rock surfaces, polymer viscosity as a function of shear rate, polymer and electrolytes concentrations, permeability reduction, and inaccessible pore volume. A time-splitting algorithm is used to "independently" solve advection, diffusion/dispersion, and chemical reactions.
2010, 5(3): 603-615 doi: 10.3934/nhm.2010.5.603 +[Abstract](1630) +[PDF](545.4KB)
Abstract:
We present here some results provided by a multiscale resolution method using both Finite Volumes and Finite Elements. This method is aimed at solving very large diffusion problems with highly oscillating coefficients. As an illustrative example, we simulate models of cement media, where very strong variations of diffusivity occur. As a by-product of our simulations, we compute the effective diffusivities of these media. After a short introduction, we present a theorical description of our method. Numerical experiments on a two dimensional cement paste are presented subsequently. The third section describes the implementation of our method in the calculus code MPCube and its application to a sample of mortar. Finally, we discuss strengths and weaknesses of our method, and present our future works on this topic.
2010, 5(3): 617-633 doi: 10.3934/nhm.2010.5.617 +[Abstract](2411) +[PDF](688.0KB)
Abstract:
We characterize the vanishing viscosity limit for multi-dimensional conservation laws of the form

$u_t +$div$\mathfrak{f}(x,u)=0, \quad u|_{t=0}=u_0$

in the domain $\mathbb R^+\times\mathbb R^N$. The flux $\mathfrak{f}=\mathfrak{f}(x,u)$ is assumed locally Lipschitz continuous in the unknown $u$ and piecewise constant in the space variable $x$; the discontinuities of $\mathfrak{f}(\cdot,u)$ are contained in the union of a locally finite number of sufficiently smooth hypersurfaces of $\mathbb R^N$. We define "$\mathcal G_{VV}$-entropy solutions'' (this formulation is a particular case of the one of [3]); the definition readily implies the uniqueness and the $L^1$ contraction principle for the $\mathcal G_{VV}$-entropy solutions. Our formulation is compatible with the standard vanishing viscosity approximation

$u^\varepsilon_t +$div$(\mathfrak{f}(x,u^\varepsilon)) =\varepsilon \Delta u^\varepsilon, \quad u^\varepsilon|_{t=0}=u_0, \quad \varepsilon\downarrow 0,$

of the conservation law. We show that, provided $u^\varepsilon$ enjoys an $\varepsilon$-uniform $L^\infty$ bound and the flux $\mathfrak{f}(x,\cdot)$ is non-degenerately nonlinear, vanishing viscosity approximations $u^\varepsilon$ converge as $\varepsilon \downarrow 0$ to the unique $\mathcal G_{VV}$-entropy solution of the conservation law with discontinuous flux.

2010, 5(3): 635-647 doi: 10.3934/nhm.2010.5.635 +[Abstract](1740) +[PDF](225.1KB)
Abstract:
We consider a simplified model for two-phase flows in one- dimensional heterogeneous porous media made of two different rocks. We focus on the effects induced by the discontinuity of the capillarity field at interface. We first consider a model with capillarity forces within the rocks, stating an existence/uniqueness result. Then we look for the asymptotic problem for vanishing capillarity within the rocks, remaining only on the interface. We show that either the solution to the asymptotic problem is the optimal entropy solution to a scalar conservation law with discontinuous flux, or it admits a non-classical shock at the interface modeling oil-trapping.
2010, 5(3): 649-660 doi: 10.3934/nhm.2010.5.649 +[Abstract](2178) +[PDF](222.9KB)
Abstract:
We present in this paper a review of some recent works dedicated to the numerical interfacial coupling of fluid models. One main motivation of the whole approach is to provide some meaningful methods and tools in order to compute unsteady patterns, while using distinct existing CFD codes in the nuclear industry. Thus, the main objective is to derive suitable boundary conditions for the codes to be coupled. A first section is devoted to a review of some attempts to couple: (i) 1D and 3D codes, (ii) distinct homogeneous two-phase flow models, (iii) fluid and porous models. More details on numerical procedures described in this section can be found in companion papers. Then we detail in a second section a way to couple a two-fluid hyperbolic model and an homogeneous relaxation model.
2010, 5(3): 661-674 doi: 10.3934/nhm.2010.5.661 +[Abstract](2110) +[PDF](547.1KB)
Abstract:
We consider weak solutions of hyperbolic conservation laws as singular limits of solutions for associated complex regularized problems. We are interested in situations such that undercompressive (Non-Laxian) shock waves occur in the limit. In this setting one can view the conservation law as a macroscale formulation while the regularization can be understood as the microscale model.
With this point of view it appears natural to solve the macroscale model by a heterogeneous multiscale approach in the sense of E&Engquist[7]. We introduce a new mass-conserving numerical method based on this concept and test it on scalar model problems. This includes applications from phase transition theory as well as from two-phase flow in porous media.

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