# American Institute of Mathematical Sciences

ISSN:
1556-1801

eISSN:
1556-181X

All Issues

## Networks & Heterogeneous Media

March 2016 , Volume 11 , Issue 1

Special issue on contemporary topics in conservation laws

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2016, 11(1): i-ii doi: 10.3934/nhm.2016.11.1i +[Abstract](1655) +[PDF](105.3KB)
Abstract:
During last 20 years the theory of Conservation Laws underwent a dramatic developmen. Networks and Heterogeneous Media is dedicating two consecutive Special Issues to this topic. Researchers belonging to some of the major schools in this subject contribute to these two issues, offering a view on the current state of the art, as well pointing to new research themes within areas already exposed to more traditional methodologies.

2016, 11(1): 1-27 doi: 10.3934/nhm.2016.11.1 +[Abstract](2319) +[PDF](2625.2KB)
Abstract:
We propose a variational framework for the resolution of a non-hydrostatic Saint-Venant type model with bottom topography. This model is a shallow water type approximation of the incompressible Euler system with free surface and slightly differs from the Green-Nagdhi model, see [13] for more details about the model derivation.
The numerical approximation relies on a prediction-correction type scheme initially introduced by Chorin-Temam [17] to treat the incompressibility in the Navier-Stokes equations. The hyperbolic part of the system is approximated using a kinetic finite volume solver and the correction step implies to solve a mixed problem where the velocity and the pressure are defined in compatible finite element spaces.
The resolution of the incompressibility constraint leads to an elliptic problem involving the non-hydrostatic part of the pressure. This step uses a variational formulation of a shallow water version of the incompressibility condition.
Several numerical experiments are performed to confirm the relevance of our approach.
2016, 11(1): 29-47 doi: 10.3934/nhm.2016.11.29 +[Abstract](2063) +[PDF](866.7KB)
Abstract:
We revisit the entropy formulation and the wave-front tracking construction of physically admissible solutions of the Aw-Rascle and Zhang (ARZ) second-order'' model for vehicular traffic. A Kruzhkov-like family of entropies is introduced to select the admissible shocks. This tool allows to define rigorously the appropriate notion of admissible weak solution and to approximate the solutions of the ARZ model with point constraint. Stability of solutions w.r.t. strong convergence is justified. We propose a finite volumes numerical scheme for the constrained ARZ, and we show that it can correctly locate contact discontinuities and take the constraint into account.
2016, 11(1): 49-67 doi: 10.3934/nhm.2016.11.49 +[Abstract](2192) +[PDF](4448.8KB)
Abstract:
This paper is devoted to the overview of recent results concerning nonlocal systems of conservation laws. First, we present a predator -- prey model and, second, a model for the laser cutting of metals. In both cases, these equations lead to interesting pattern formation.
2016, 11(1): 69-87 doi: 10.3934/nhm.2016.11.69 +[Abstract](2292) +[PDF](969.8KB)
Abstract:
We performed numerical simulations of blood flow in arteries with a variable stiffness and cross-section at rest using a finite volume method coupled with a hydrostatic reconstruction of the variables at the interface of each mesh cell. The method was then validated on examples taken from the literature. Asymptotic solutions were computed to highlight the effect of the viscous and viscoelastic source terms. Finally, the blood flow was computed in an artery where the cross-section at rest and the stiffness were varying. In each test case, the hydrostatic reconstruction showed good results where other simpler schemes did not, generating spurious oscillations and nonphysical velocities.
2016, 11(1): 89-105 doi: 10.3934/nhm.2016.11.89 +[Abstract](1809) +[PDF](444.8KB)
Abstract:
We consider the boundary value problem for the phase transition (PT) model, introduced in [4] and in [7]. By using the wave-front tracking technique, we prove existence of solutions when the initial and boundary conditions have finite total variation.
2016, 11(1): 107-121 doi: 10.3934/nhm.2016.11.107 +[Abstract](2665) +[PDF](628.6KB)
Abstract:
We consider an extension of the traffic flow model proposed by Lighthill, Whitham and Richards, in which the mean velocity depends on a weighted mean of the downstream traffic density. We prove well-posedness and a regularity result for entropy weak solutions of the corresponding Cauchy problem, and use a finite volume central scheme to compute approximate solutions. We perform numerical tests to illustrate the theoretical results and to investigate the limit as the convolution kernel tends to a Dirac delta function.
2016, 11(1): 123-143 doi: 10.3934/nhm.2016.11.123 +[Abstract](2213) +[PDF](419.9KB)
Abstract:
The Escalator Boxcar Train method (EBT) is a numerical method for structured population models of McKendrick -- von Foerster type. Those models consist of a certain class of hyperbolic partial differential equations and describe time evolution of the distribution density of the structure variable describing a feature of individuals in the population. The method was introduced in late eighties and widely used in theoretical biology, but its convergence was proven only in recent years using the framework of measure-valued solutions. Till now the EBT method was developed only for scalar equation models. In this paper we derive a full numerical EBT scheme for age-structured, two-sex population model (Fredrickson-Hoppensteadt model), which consists of three coupled hyperbolic partial differential equations with nonlocal boundary conditions. It is the first step towards extending the EBT method to systems of structured population equations.
2016, 11(1): 145-162 doi: 10.3934/nhm.2016.11.145 +[Abstract](1946) +[PDF](1316.9KB)
Abstract:
We describe a shock-capturing streamline diffusion space-time discontinuous Galerkin (DG) method to discretize the shallow water equations with variable bottom topography. This method, based on the entropy variables as degrees of freedom, is shown to be energy stable as well as well-balanced with respect to the lake at rest steady state. We present numerical experiments illustrating the numerical method.
2016, 11(1): 163-180 doi: 10.3934/nhm.2016.11.163 +[Abstract](1774) +[PDF](2414.6KB)
Abstract:
The nonlocal nonlinear aggregation equation in one space dimension is investigated. In the so-called attractive case smooth solutions blow up in finite time, so that weak measure solutions are introduced. The velocity involved in the equation becomes discontinuous, and a particular care has to be paid to its definition as well as the formulation of the corresponding flux. When this is done, the notion of duality solutions allows to obtain global in time existence and uniqueness for measure solutions. An upwind finite volume scheme is also analyzed, and the convergence towards the unique solution is proved. Numerical examples show the dynamics of the solutions after the blow up time.
2016, 11(1): 181-201 doi: 10.3934/nhm.2016.11.181 +[Abstract](2369) +[PDF](465.8KB)
Abstract:
We investigate a Keller-Segel model with quorum sensing and a fractional diffusion operator. This model describes the collective cell movement due to chemical sensing with flux limitation for high cell densities and with anomalous media represented by a nonlinear, degenerate fractional diffusion operator. The purpose of this paper is to introduce and prove the existence of a properly defined entropy solution.

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