We propose a general framework for the study of L 1 contractive semigroups solutions to conservation laws with discontinuous flux: $$ u_t + \mathfrak{f}(x,u)_x=0, \qquad \mathfrak{f}(x,u)= \left\{\begin{array}{ll} f^l(u),& x < 0,\\ f^r(u), & > 0, \end{array} \right.\quad\quad\quad (\rm CL) where fluxes f l , r are mainly assumed be continuous. Developing ideas number preceding works (Baiti and Jenssen in J Differ Equ 140(1):161–185, 1997; Towers SIAM Numer Anal 38(2):681–698, 2000;...

Abstract Discrete duality finite volume schemes on general meshes, introduced by Hermeline and Domelevo Omnès for the Laplace equation, are proposed nonlinear diffusion problems in 2D with nonhomogeneous Dirichlet boundary condition. This approach allows discretization of non linear fluxes such a way that discrete operator inherits key properties continuous one. Furthermore, it is well adapted to very meshes including case nonconformal locally refined meshes. We show approximate solution...

The main goal of this paper is to propose a convergent finite volume method for reaction–diffusion system with cross-diffusion. First, we sketch an existence proof class cross-diffusion systems. Then the standard two-point fluxes are used in combination nonlinear positivity-preserving approximation coefficients. Existence and uniqueness approximate solution addressed, it also shown that scheme converges corresponding weak studied model. Furthermore, provide stability analysis study...

We consider a class of doubly nonlinear degenerate hyperbolic-parabolic equations with homogeneous Dirichlet boundary conditions, for which we first establish the existence and uniqueness entropy solutions. then turn to construction analysis discrete duality finite volume schemes (in spirit Domelevo Omnès [43]) these problems in two three spatial dimensions. derive series formulas dissipation inequalities schemes. solutions problems, prove that sequences approximate generated by converge...

We characterize the vanishing viscosity limit for multi-dimensionalconservation laws of form$ u_t + $div$ \mathfrak{f}(x,u)=0, \quad u|_{t=0}=u_0 $in domain $\mathbb R^+\times\mathbb R^N$. The flux $\mathfrak{f}=\mathfrak{f}(x,u)$ is assumed locallyLipschitz continuous in unknown $u$ and piecewise constant space variable $x$; discontinuities of$\mathfrak{f}(\cdot,u)$ are contained union a locally finite number sufficiently smoothhypersurfaces define '$\mathcal G_{VV}$-entropy solutions''...

In this paper we model pedestrian flows evacuating a narrow corridor through an exit by one-dimensional hyperbolic conservation law with point constraint in the spirit of [Colombo and Goatin, J. Differential Equations, 2007]. We introduce nonlocal to restrict flux at maximum value p(ξ), where ξ is weighted averaged instantaneous density crowd upstream vicinity exit. Choosing non-increasing function p(⋅), are able capacity drop phenomenon Existence stability results for Cauchy problem...

In this paper we investigate well-posedness for the problem<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript t Baseline plus d i v phi left-parenthesis u right-parenthesis equals f"><mml:semantics><mml:mrow><mml:msub><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mi>div</mml:mi><mml:mo>⁡</mml:mo><mml:mi>φ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo...

In this paper we investigate numerically the model for pedestrian traffic proposed in [B.Andreianov, C.Donadello, M.D.Rosini, Crowd dynamics and conservation laws with nonlocal constraints capacity drop, Mathematical Models Methods Applied Sciences 24 (13) (2014) 2685-2722]. We prove convergence of a scheme based on constraint finite volume method validate it an explicit solution obtained above reference. then perform ad hoc simulations to qualitatively under consideration by proving its...

Journal Article On 3D DDFV discretization of gradient and divergence operators. I. Meshing, operators discrete duality Get access Boris Andreianov, Andreianov * Laboratoire de Mathématiques CNRS UMR 6623, Université Franche-Comté, 16 route Gray, 25030 Besançon Cedex, France *Corresponding author: boris.andreianov@univ-fcomte.fr Search for other works by this author on: Oxford Academic Google Scholar Mostafa Bendahmane, Bendahmane Institut Bordeaux, Victor Segalen, 33076 Bordeaux France,...

We provide a complete study of the model investigated in [Coclite, Garavello, SIAM J. Math. Anal., 2010]. prove well-posedness solutions obtained as vanishing viscosity limits for Cauchy problem scalar conservation laws $ ρ_{h, t} + f_h(ρ_h)_x = 0$, $h∈ \{1, ..., m+n\}$, on junction where $m$ incoming and $n$ outgoing edges meet. Our analysis definition admissible solution rely upon description set edge-wise constant its properties, which is some interest own. The Riemann solver at...

Conservation laws of the form ∂ t u + x f(x;u) = 0 with space-discontinuous flux f(x;⋅) f l (⋅)1 x&lt;0 r x&gt;0 were deeply investigated in past ten years, a particular emphasis case where fluxes are "bell-shaped". In this paper, we introduce and exploit idea transmission maps for interface condition at discontinuity, leading to well-posedness Cauchy problem general shape l,r . The design convergence monotone Finite Volume schemes based on one-sided approximate Riemann solvers then...

Abstract. We present a detailed survey of discrete functional analysis tools (consistency results, Poincaré and Sobolev embedding inequalities, W 1, p compactness, compactness in space time) for the so-called Discrete Duality Finite Volume (DDFV) schemes three dimensions. concentrate mainly on 3D CeVe-DDFV scheme presented [IMA J. Numer. Anal., 32 (2012), pp. 1574–1603]. Some our results are new, such as general time-compactness result based upon idea Kruzhkov (1969); others generalize ideas...

Hyperbolic conservation laws of the form ut + div f(t, x;u) = 0 with discontinuous in (t,x) flux function f attracted much attention last 20 years, because difficulties adaptation classical Kruzhkov approach developed for smooth case. In discontinuous-flux case, non-uniqueness mathematically consistent admissibility criteria results infinitely many different notions solution. A way to describe all resulting L1-contractive solvers within a unified was proposed work [Andreianov, Karlsen,...