Mazen Saad

Finite volume methods for degenerate chemotaxis model

A finite volume method for solving the degenerate chemotaxis model is presented, along with numerical examples. This model consists of a degenerate parabolic convection-diffusion PDE for the density of the cell-population coupled to a parabolic PDE for the chemoattractant concentration. It is shown that discrete solutions exist, and the scheme converges.

Study of Full Implicit Petroleum Engineering Finite-Volume Scheme for Compressible Two-Phase Flow in Porous Media

An industrial scheme, to simulate the compressible two-phase flow in porous media, consists of a finite volume method together with a phase-by-phase upstream scheme. The implicit finite volume scheme satisfies industrial constraints of robustness since the proposed scheme discretizes the equations with gravity and capillary terms. We show that the proposed scheme satisfies the maximum principle for the saturation, a discrete-energy estimate on the pressures, and a function of the saturation that denotes capillary terms. These stability results allow us to derive the convergence of a subsequence to a weak solution of the continuous equations as the size of the discretization tends to zero. To our knowledge, this is the first convergence result of a finite volume scheme in the case of two-phase compressible flow in several space dimensions. The proof is given for the complete system when the density of each phase depends on its own pressure.

On the efficacy of a control volume finite element method for the capture of patterns for a volume-filling chemotaxis model

In this paper, a control volume finite element scheme for the capture of spatial patterns for a volume-filling chemotaxis model is proposed and analyzed. The diffusion term, which generally involves an anisotropic and heterogeneous diffusion tensor, is discretized by piecewise linear conforming triangular finite elements (P1-FEM). The other terms are discretized by means of an upstream finite volume scheme on a dual mesh, where the dual volumes are constructed around the vertices of each element of the original mesh. The scheme ensures the validity of the discrete maximum principle under the assumption that the transmissibility coefficients are nonnegative. The convergence analysis is based on the establishment of a priori estimates on the cell density, these estimates lead to some compactness arguments in L 2 based on the use of the Kolmogorov compactness theorem. Finally, we show some numerical results to illustrate the effectiveness of the scheme to capture the pattern formation for the mathematical model.

A combined finite volume---nonconforming finite element scheme for compressible two phase flow in porous media

We propose and analyze a combined finite volume–nonconforming finite element scheme on general meshes to simulate the two compressible phase flow in porous media. The diffusion term, which can be anisotropic and heterogeneous, is discretized by piecewise linear nonconforming triangular finite elements. The other terms are discretized by means of a cell-centered finite volume scheme on a dual mesh, where the dual volumes are constructed around the sides of the original mesh. The relative permeability of each phase is decentred according the sign of the velocity at the dual interface. This technique also ensures the validity of the discrete maximum principle for the saturation under a non restrictive shape regularity of the space mesh and the positiveness of all transmissibilities. Next, a priori estimates on the pressures and a function of the saturation that denote capillary terms are established. These stabilities results lead to some compactness arguments based on the use of the Kolmogorov compactness theorem, and allow us to derive the convergence of a subsequence of the sequence of approximate solutions to a weak solution of the continuous equations, provided the mesh size tends to zero. The proof is given for the complete system when the density of the each phase depends on its own pressure.

A predator-prey system with L1 data

In this paper, we are concerned with a system of nonlinear partial differential equations modeling a predator–prey system in heterogeneous habitats. Preys are assumed to follow a logistic growth in the absence of predation, and predators are assumed to feed on preys with a Holling type II functional response to prey density. Also, interactions between predators are modelized by the statement of a food pyramid condition. Assuming no-flux boundary conditions and L 1 data, we prove the existence of at least one weak solution.

A coupled anisotropic chemotaxis-fluid model

In this article, the mathematical analysis of a model arising from biology consisting of diffusion, chemotaxis with volume filling effect and transport through an incompressible fluid, is studied. Motivated by numerical and modeling issues, the global-in-time existence of weak solutions to this model is investigated. The novelty with respect to other related papers lies in the presence of two-sidedly nonlinear degenerate diffusion and of anisotropic and heterogeneous diffusion tensors where we prove the global existence for a Chemotaxis-Navier-Stokes system in space dimensions less than or equal to four and we show the uniqueness of weak solutions for the Chemotaxis-Stokes system in two or three space dimensions under further assumptions.

Original article: Degenerate two-phase compressible immiscible flow in porous media: The case where the density of each phase depends on its own pressure

High-resolution color and monochrome video images are recorded on and reproduced from a record medium. Red, green and blue signal components are recorded as respective monochrome-like signals in separate tracks, and since a chrominance component of the video signal is not recorded in each track, the bandwidth of the recorded video signal may exceed the bandwidth of a video signal that normally is stored on the record medium. The high-resolution color video signal is reproduced by reproducing three frames which contain the red, green and blue signal components, respectively, and by supplying these signal components as a single high-resolution color image. A high-resolution monochrome video signal is stored by sampling the monochrome video signal at a given clock signal to produce a first sampled signal and by sampling the monochrome video signal at an inverse clock signal to produce a second sampled signal which are recorded as separate monochrome frames. Upon reproduction, two frames corresponding to the two sampled signals are reproduced and combined in accordance with the clock and inverse clock signals to produce a high-resolution monochrome video image.

Analysis of a finite volume method for a bone growth system in vivo

This paper is devoted to the convergence analysis of a finite volume method and numerical simulations of a reaction-cross diffusion system arising from a bone growth model. This model describes the evolution of mesenchymal stem cells, osteoblasts, bone matrix and osteogenic growth factor. We propose a numerical scheme based on an implicit finite volume method constructed on an orthogonal mesh. Lack of the regularity of the approximate system is overcome by stability results which allow to obtain estimates on the translates, apply the Kolmogorov theorem in order to get compactness and show the convergence of the proposed scheme. The efficiency and robustness of the scheme are shown in simulating a situation in bone growth: the healing of a skull fracture in rats.

Front tracking for two‐phase flow in reservoir simulation by adaptive mesh

In this article, an algorithm for the numerical approximation of two-phase flow in porous media by adaptive mesh is presented. A convergent and conservative finite volume scheme for an elliptic equation is proposed, together with the finite difference schemes, upwind and MUSCL, for a hyperbolic equation on grids with local refinement. Hence, an IMPES method is applied in an adaptive composite grid to track the front of a moving solution. An object-oriented programmation technique is used. The computational results for different examples illustrate the efficiency of the proposed algorithm. c

Analysis of a positive CVFE scheme for simulating breast cancer development, local treatment and recurrence

In this paper, a positive CVFE scheme for simulating an anisotropic breast cancer development is analyzed. The mathematical model includes reaction–diffusion-convection terms with an anisotropic heterogeneous diffusion tensor. The diffusion term is discretized using a finite element method combined with the use of Godunov scheme over a primal triangular mesh. The convective term is discretized using a nonclassical upwind finite volume scheme over a barycentric dual mesh. The scheme ensures the validity of the discrete positivity preserving and other discrete properties without any restriction on the transmissibility coefficients. Finally, a numerical simulation is provided to simulate the spread of tumor cells before and after applying a local treatment using the surgery.