Showing papers in "Mathematical Models and Methods in Applied Sciences in 2000"
TL;DR: A discrete velocity model of this equation is proposed using the minimum entropy principle to define a discrete equilibrium function, and this model ensures positivity of solutions, conservation of moments, and dissipation of entropy.
Abstract: We present a numerical method for computing transitional flows as described by the BGK equation of gas kinetic theory. Using the minimum entropy principle to define a discrete equilibrium function, a discrete velocity model of this equation is proposed. This model, like the continuous one, ensures positivity of solutions, conservation of moments, and dissipation of entropy. The discrete velocity model is then discretized in space and time by an explicit finite volume scheme which is proved to satisfy the previous properties. A linearized implicit scheme is then derived to efficiently compute steady-states; this method is then verified with several test cases.
232 citations
TL;DR: The existence of maximum entropy solutions for a wide class of reduced moment problems on arbitrary open subsets of ℝd is considered in this article, where a precise condition is presented under which solvability of the moment problem implies existence of a maximum entropy solution.
Abstract: The existence of maximum entropy solutions for a wide class of reduced moment problems on arbitrary open subsets of ℝd is considered. In particular, new results for the case of unbounded domains are obtained. A precise condition is presented under which solvability of the moment problem implies existence of a maximum entropy solution.
135 citations
TL;DR: In this paper, the boundary conditions for the large eddy model were modified from strict adherence to slip with resistance, where the resistance coefficient is a function of the averaging radius, so that the model's boundary conditions reduce to a no-slip as the average radius decreases to zero.
Abstract: We present two modifications of continuum models used in large eddy simulation. The first modification is a closure approximation which better attenuates small eddies. The second modification is a change in the boundary conditions for the large eddy model from strict adherence to slip with resistance. For model consistency, the resistance coefficient is a function of the averaging radius so that the model's boundary conditions reduce to a no-slip as the averaging radius decreases to zero.
122 citations
TL;DR: A multicell model to describe the evolution of tumour growth from the avascular stage to the vascular one through the angiogenic process is proposed and deduced both in a continuum mechanics framework and by a lattice scheme in order to put in evidence the relation between microscopic phenomena and macroscopic parameters.
Abstract: This paper proposes a multicell model to describe the evolution of tumour growth from the avascular stage to the vascular one through the angiogenic process. The model is able to predict the formation of necrotic regions, the control of mitosis by the presence of an inhibitory factor, the angiogenesis process with proliferation of capillaries just outside the tumour surface with penetration of capillary sprouts inside the tumour, the regression of the capillary network induced by the tumour when angiogenesis is controlled or inhibited, say as an effect of angiostatins, and finally the regression of the tumour size. The three-dimensional model is deduced both in a continuum mechanics framework and by a lattice scheme in order to put in evidence the relation between microscopic phenomena and macroscopic parameters. The evolution problem can be written as a free-boundary problem of mixed hyperbolic–parabolic type coupled with an initial-boundary value problem in a fixed domain.
99 citations
TL;DR: In this paper, the stability of the Vlasov-Poisson-Fokker-Planck with respect to the variation of its constant parameters, including the thermal velocity and the scaled thermal mean free path, is analyzed.
Abstract: In this paper the stability of the Vlasov–Poisson–Fokker–Planck with respect to the variation of its constant parameters, the scaled thermal velocity and the scaled thermal mean free path, is analy...
89 citations
TL;DR: In this article, a new class of particle methods for conservation laws, based on numerical flux functions to model the interactions between moving particles, is presented. But the method is not suitable for the one-dimensional shock tube problem.
Abstract: We derive a new class of particle methods for conservation laws, which are based on numerical flux functions to model the interactions between moving particles. The derivation is similar to that of classical finite-volume methods; except that the fixed spatial mesh in a finite-volume method is substituted by so-called mass packets of particles. We give some numerical results on a shock wave solution for Burgers equation as well as the well-known one-dimensional shock tube problem.
73 citations
TL;DR: In this paper, the convergence and consistency of approximate solutions derived by the modified Godunov scheme for the initial-boundary value problem to a bipolar hydrodynamic model for semiconductors are proved using the compensated compactness framework.
Abstract: The convergence and consistency of approximate solutions derived by the modified Godunov scheme for the initial–boundary value problem to a bipolar hydrodynamic model for semiconductors are proved using the compensated compactness framework. The information of weak solutions to satisfy the boundary conditions is also displayed. The zero relaxation limit of the bipolar hydrodynamic model towards the drift-diffusion model is carried out when the current relaxation time tends to zero.
70 citations
TL;DR: In this paper, the authors presented a rigorous asymptotic analysis of the two-dimensional Vlasov equation when the magnetic field tends to infinity and the observation time scale increases accordingly.
Abstract: When charged particles are submitted to a large external magnetic field, their movement in first approximation occurs along the magnetic field lines and obeys a one-dimensional Vlasov equation along these field lines. However, when observing the particles on a sufficiently long time scale, a drift phenomenon perpendicular to the magnetic field lines superposes to this first movement. In this paper, we present a rigorous asymptotic analysis of the two-dimensional Vlasov equation when the magnetic field tends to infinity, the observation time scale increases accordingly. Techniques based on the two-scale convergence and the introduction of a second problem enable us to find an equation verified by the weak limit of the distribution function.
61 citations
TL;DR: In this article, the authors studied the relaxation limit of the 3-D bipolar hydrodynamic model for semiconductors and proved the convergence of the weak solutions to the bipolar Euler-Poisson system towards the solutions of the bipolar drifthyphen;diffusion system, as the relaxation time tends to zero.
Abstract: The aim of this paper is the study of the relaxation limit of the 3-D bipolar hydrodynamic model for semiconductors. We prove the convergence for the weak solutions to the bipolar Euler–Poisson system towards the solutions to the bipolar drifthyphen;diffusion system, as the relaxation time tends to zero.
60 citations
TL;DR: In this paper, it was shown that a reaction-diffusion inclusion provides a sub-optimal approximation for anisotropic motion by mean curvature in the nonsmooth case.
Abstract: We prove that a reaction-diffusion inclusion provides a sub-optimal approximation for anisotropic motion by mean curvature in the nonsmooth case. This result is valid in any space dimension and with a time-dependent driving force, provided we assume the existence of a regular flow. The crystalline case is included. As a by-product of our analysis, a comparison theorem between regular flows is obtained. This result implies uniqueness of the original flow.
53 citations
TL;DR: In this article, the authors consider the homogenization of sequences of integral functionals defined on media with several length-scales and show that the macroscopic behavior of this structure underlines an effective energy density which is lower than that of the best possible multirank laminate.
Abstract: We consider the homogenization of sequences of integral functionals defined on media with several length-scales. Our general results connected to the corresponding homogenized functional are used to analyze new types of structures and to illustrate the wide range of effective properties achievable through reiteration. In particular, we consider a two-phase structure which is optimal when the integrand is a quadratic form and point out examples where the macroscopic behavior of this structure underlines an effective energy density which is lower than that of the best possible multirank laminate. We also present some results connected to a reiterated structure where the effective property is extremely sensitive of the growth of the integrand.
TL;DR: In this paper, the authors prove the existence of weak solutions for mathematical models of miscible and immiscible flows through porous medium, which does not allow us to use classical variational formulations of the equations.
Abstract: In this paper, we prove the existence of weak solutions for mathematical models of miscible and immiscible flows through porous medium. An important difficulty comes from the modelization of the wells, which does not allow us to use classical variational formulations of the equations.
TL;DR: In this paper, a nonconforming mixed finite element scheme for the approximate solution of the time-harmonic Maxwell's equations in a three-dimensional, bounded domain with absorbing boundary conditions on artificial boundaries is presented.
Abstract: We present a nonconforming mixed finite element scheme for the approximate solution of the time-harmonic Maxwell's equations in a three-dimensional, bounded domain with absorbing boundary conditions on artificial boundaries. The numerical procedures are employed to solve the direct problem in magnetotellurics consisting in determining a scattered electromagnetic field in a model of the earth having bounded conductivity anomalies of arbitrary shapes. A domain-decomposition iterative algorithm which is naturally parallelizable and is based on a hybridization of the mixed method allows the solution of large three-dimensional models. Convergence of the approximation by the mixed method is proved, as well as the convergence of the iteration.
TL;DR: In this article, the unsteady flow of a viscoelastic fluid in a straight, long, rigid pipe, driven by a suddenly imposed pressure gradient is studied, where the used model is the Oldroyd-B fluid modified with the use of a nonconstant viscosity, which includes the effect of the shear-thinning of many fluids.
Abstract: The unsteady flow of a viscoelastic fluid in a straight, long, rigid pipe, driven by a suddenly imposed pressure gradient is studied. The used model is the Oldroyd-B fluid modified with the use of a nonconstant viscosity, which includes the effect of the shear-thinning of many fluids. The main application considered is in blood flow. Two coupled nonlinear equations are solved by a spectral collocation method in space and the implicit trapezoidal finite difference method in time. The presented results show the role of the non-Newtonian terms in unsteady phenomena.
TL;DR: In this paper, the analysis of two Helmholtz equations in ℝ2 coupled via quasiperiodic transmission conditions on a set of piecewise smooth interfaces is devoted to the analysis.
Abstract: This paper is devoted to the analysis of two Helmholtz equations in ℝ2 coupled via quasiperiodic transmission conditions on a set of piecewise smooth interfaces. The solution of this system is quasiperiodic in one direction and satisfies outgoing wave conditions with respect to the other direction. It is shown that Maxwell's equations for the diffraction of a time-harmonic oblique incident plane wave by periodic interfaces can be reduced to problems of this kind. The analysis is based on a strongly elliptic variational formulation of the differential problem in a bounded periodic cell involving nonlocal boundary operators. We obtain existence and uniqueness results for solutions corresponding to electromagnetic fields with locally finite energy. Special attention is paid to the regularity and leading asymptotics of solutions near the edges of the interface.
TL;DR: In this paper, numerical solutions of recent hydrodynamical models of semiconductors are computed in one-space dimension, where two models are taken into consideration: the first one has been developed by Blotekjaer, Baccarani et al., and the second one by Anile et al.
Abstract: Numerical solutions of recent hydrodynamical models of semiconductors are computed in one-space dimension. Such models describe charge transport in semiconductor devices. Two models are taken into consideration. The first one has been developed by Blotekjaer, Baccarani et al., and the second one by Anile et al. In both cases the system of equations can be written as a convection-diffusion type system, with a right-hand side describing relaxation effects and interaction with a self-consistent electric field. The numerical scheme is a splitting scheme based on the Nessyahu–Tadmor scheme for the hyperbolic step, and a semi-implicit scheme for the relaxation step. The numerical results are compared to detailed Monte-Carlo simulation.
TL;DR: In this paper, a novel construction of wavelets on the unit interval is introduced, and explicit upper bounds for the length of the modified border wavelets filters can be given, which insures a good localization of the border wavelet when a triangular biorthogonalization scheme is employed.
Abstract: This paper introduces a novel construction of wavelets on the unit interval. With this construction, explicit upper bounds for the length of the modified border wavelets filters can be given. This insures a good localization of the border wavelets when a triangular biorthogonalization scheme is employed. The resulting wavelet bases are then well-suited for the adaptive solution of partial differential equations.
TL;DR: In this article, the dispersion limit of the Schrodinger equation with Sobolev regularity was studied, and the strict hyperbolicity and genuine nonlinearity were proved.
Abstract: We study the semiclassical limit of the general derivative nonlinear Schrodinger equation for initial data with Sobolev regularity, before shocks appear in the limit system. The strict hyperbolicity and genuine nonlinearity is proved for the dispersion limit of the derivative nonlinear Schrodinger equation.
TL;DR: In this paper, it was shown that the motion of the particles can be approximatively decomposed as the sum of a fast rotation on Larmor circles, an advection along the magnetic lines and a small drift orthogonal to both electric and magnetic fields.
Abstract: Consider a plasma in a strong constant magnetic field with self-consistent electric field. We present here the formal derivation that leads to the so-called guiding center approximation, and justify it in the case of a well-prepared initial density of particles. More precisely, we prove that the motion of the particles can be approximatively decomposed as the sum of a fast rotation on Larmor circles, an advection along the magnetic lines and a small drift orthogonal to both electric and magnetic fields.
TL;DR: In this paper, the authors studied the global existence and the time asymptotic behavior of solutions to the model system with discontinuous initial data and showed that the solutions approach the corresponding smooth traveling waves with the rate t-1/4 in the supremum norm.
Abstract: A one-dimensional model of radiating gas is obtained from approximating the system of radiating gas with thermo-nonequilibrium. The model system consists of a conservation law and a linear elliptic equation. In this paper, we study the global existence and the time asymptotic behavior of solutions to the model system with discontinuous initial data. Since the first equation is hyperbolic, the solutions contain discontinuities for any positive time. But, the uniqueness of solutions in weak sense holds by imposing the entropy condition. The main concern of this research is to investigate the behavior of the discontinuities contained in the solutions. It is proved that the set of discontinuous points consists of a certain C1-curve. This discrepancy of values at the discontinuities of the solutions is shown to decay to zero exponentially fast as time tends to infinity. This property is utilized in showing that the solutions approach the corresponding smooth traveling waves with the rate t-1/4 in the supremum norm.
TL;DR: It is proved that, for a linear finite element space, the bounds converge to the exact result at the optimal rate: for sufficiently regular data and output functionals, where H is the diameter of the mesh which provides the solution and adjoint required by the technique.
Abstract: We present the numerical analysis of a new a posteriori finite element procedure that provides rigorous, constant-free, lower and upper bounds for linear functional outputs of coercive second-order partial differential equations. We prove that, for a linear finite element space, the bounds converge to the exact result at the optimal rate: for sufficiently regular data and output functionals, where H is the diameter of the mesh which provides the solution and adjoint required by our technique.
TL;DR: In this paper, it was shown that when time goes to infinity, the rescaled self-consistent potential can be identified as the Coulomb potential, and that the Schrodinger-Poisson and Wigner-Wigner systems are asymptotically simplified.
Abstract: Using an appropriate scaling group for the 3-D Schrodinger–Poisson equation and the equivalence between the Schrodinger formalism and the Wigner representation of quantum mechanics it is proved that, when time goes to infinity, the limit of the rescaled self-consistent potential can be identified as the Coulomb potential. As a consequence, Schrodinger–Poisson and Wigner–Poisson systems are asymptotically simplified and their long-time behavior is explained through the solutions of the corresponding linear limit problems.
TL;DR: In this paper, the stability properties of a sandwich beam consisting of two outer layers and a thin core were examined. And they showed for both clamped and hinged boundary conditions that if rotational inertia terms are neglected, the model is described by an analytic semigroup.
Abstract: We examine the stability properties of a sandwich beam consisting of two outer layers and a thin core. The outer layers are modeled as Euler Bernoulli beams and the inner core provides both elastic and viscous resistance to shearing. We show for both clamped and hinged boundary conditions that (i) if rotational inertia terms are neglected, the model is described by an analytic semigroup, and (ii) if rotational inertia is retained in the outer layers, the model is uniformly exponentially stable.
TL;DR: In this article, the authors used geometrical arguments based on grain boundary symmetries to introduce crystalline interfacial energies for interfaces in polycrystalline thin films with a cubic lattice.
Abstract: We use geometrical arguments based on grain boundary symmetries to introduce crystalline interfacial energies for interfaces in polycrystalline thin films with a cubic lattice. These crystalline energies are incorporated into a multi-phase field model. Our aim is to apply the multi-phase field method to describe the evolution of faceted grain boundary triple junctions in epitaxially growing microstructures. In particular, we are interested in symmetry properties of triple junctions in tricrystalline thin films. Symmetries of triple junctions in tricrystalline films have been studied in experiments by Dahmen and Thangaraj.6,25 In accordance with their experiments, we find in numerical simulations that any two neighboring triple junctions belong to different symmetry classes. We introduce a local equilibrium condition at triple junctions which can be interpreted as a crystalline version of Young's law. The local equilibrium condition at triple junctions is purely determined by the grain boundary energies. In particular no triple junction energies are necessary to explain which triple junctions are possible. All triple junctions observed in the experiments as well as in the simulations fulfil the crystalline version of Young's law. Our approach is also capable of describing grain boundary motion in general polycrystalline thin films.
TL;DR: In this paper, a numerical procedure based on a combination of: a special refinement of the grid, a sub-element technique, a nonlinear interpolation and a split form of the integral is presented.
Abstract: An integral representation of electrograms is used for large scale simulations in an anisotropic model of the whole left ventricle. Numerical artifacts, like spurious oscillations or peaks, may appear if the computational grid is not fine enough. To avoid an excessive increase in the number of elements and nodes, we present a numerical procedure based on a combination of: a special refinement of the grid, a sub-element technique, a nonlinear interpolation and a split form of the integral. The numerical simulations show that in this way it is possible to suppress the numerical artifacts thus allowing an accurate computation of electrograms in any point inside or outside the myocardial wall.
TL;DR: In this article, the authors consider a magnetized plasma composed of electrons of low mean density and of one species of multicharged ions and derive a diffusion model for the electrons, with explicit transport coefficients which are related to the ions.
Abstract: Here we consider a magnetized plasma composed of electrons of low mean density and of one species of multicharged ions. Starting with the Vlasov–Fokker–Planck equations for both particles, we derive a diffusion model for the electrons, with explicit transport coefficients which are related to the ions. Finally, we study the kinetic boundary layers in the case of a bounded domain in space variables. For that purpose, we are led to study a Milne problem and to introduce a generalized extrapolation length.
TL;DR: In this article, the authors consider dynamical systems with a convex potential described by, where φ is a proper, convex, lower semicontinuous function and prove that these systems possess a solution whose kinetic energy is conserved through impact in the first case, or more generally, whose energy is a continuous function of time in the second case.
Abstract: We are motivated by the study of dynamical systems with a finite number of degrees of freedom, subject to unilateral convex constraints without loss of energy at impacts. If we denote the set of constraints by K, the motion is described by , where ψK is the indicatrix function of K. More generally we consider dynamical systems with a convex potential described by , where φ is a proper, convex, lower semicontinuous function. We prove that these systems possess a solution whose kinetic energy is conserved through impact in the first case, or more generally, whose energy is a continuous function of time in the second case.
TL;DR: In this article, the steady flow of a non-Newtonian fluid obeying the power law in unbounded channels and pipes is studied and the proof of existence and uniqueness of the solution for the Leray's problem for such fluid is given.
Abstract: We study the steady flow of a dilatant non-Newtonian fluid obeying the power law in unbounded channels and pipes. The proof of existence and uniqueness of the solution for the Leray's problem for such fluid is given as well as the decay estimate for the solution.
TL;DR: In this paper, the local-in-time Cauchy problem for the Schrodinger-Debye equations is studied in nonlinear optics and describes the non-resonant delayed interaction of an electromagnetic wave with a medium.
Abstract: In this paper we study the local-in-time Cauchy problem for the Schrodinger–Debye equations. This model occurs in nonlinear optics and describes the non-resonant delayed interaction of an electromagnetic wave with a medium. We extend the study to nonphysical cases such as the three-dimensional case or more general nonlinearities.
TL;DR: In this article, it was shown that Fisher's quantity of information is nonincreasing with time along solutions of the spatially homogeneous Landau equation for Maxwellian molecules, which was first seen in numerical simulation in plasma physics.
Abstract: We give a direct proof of the fact that, in any dimension of the velocity space, Fisher's quantity of information is nonincreasing with time along solutions of the spatially homogeneous Landau equation for Maxwellian molecules. This property, which was first seen in numerical simulation in plasma physics, is linked with the theory of the spatially homogeneous Boltzmann equation.