On the stability of the standard Riemann semigroup

We consider the dependence of the entropic solution of a hyperbolic system of conservation laws

on the flux function . We prove that the solution is Lipschitz continuous w.r.t. the norm of the derivative of the perturbation of . We apply this result to prove the convergence of the solution of the relativistic Euler equation to the classical limit.

     

BV estimates of Lax-Friedrichs' scheme for a class of nonlinear hyperbolic conservation laws

We give uniform BV estimates and -stability of Lax-Friedrichs' scheme for a class of systems of strictly hyperbolic conservation laws whose integral curves of the eigenvector fields are straight lines, i.e., Temple class, under the assumption of small total variation. This implies that the approximate solutions generated via the Lax-Friedrichs' scheme converge to the solution given by the method of vanishing viscosity or the Godunov scheme, and then the Glimm scheme or the wave front tracking method.

     

Delta shock waves with Dirac delta function in both components for systems of conservation laws

We study a class of non-strictly and weakly hyperbolic systems of conservation laws which contain the equations of geometrical optics as a prototype. The Riemann problems are constructively solved. The Riemann solutions include two kinds of interesting structures. One involves a cavitation where both state variables tend to zero forming a singularity, the other is a delta shock wave in which both state variables contain Dirac delta function simultaneously. The generalized Rankine–Hugoniot relation and entropy condition are proposed to solve the delta shock wave. Moreover, with the limiting viscosity approach, we show all of the existence, uniqueness and stability of solution involving the delta shock wave. The generalized Rankine–Hugoniot relation is also confirmed. Then our theory is successfully applied to two typical systems including the geometric optics equations. Finally, we present the numerical results coinciding with the theoretical analysis.     

Diffractive wave transmission in dispersive media

The aim of this paper is to study the reflection-transmission of diffractive geometrical optic rays described by semi-linear symmetric hyperbolic systems such as the Maxwell-Lorentz equations with the anharmonic model of polarization.The framework is that of P. Donnat's thesis [P. Donnat, Quelques contributions mathématiques en optique non linéaire, chapters 1 and 2, thèse, 1996] and V. Lescarret [V. Lescarret, Wave transmission in dispersive media, M3AS 17 (4) (2007) 485-535]: we consider an infinite WKB expansion of the wave over long times/distances O(1/ε) and because of the boundary, we decompose each profile into a hyperbolic (purely oscillating) part and elliptic (evanescent) part as in M. William [M. William, Boundary layers and glancing blow-up in nonlinear geometric optics, Ann. Sci. École Norm. Sup. 33 (2000) 132-209].Then to get the usual sublinear growth on the hyperbolic part of the profiles, for every corrector, we consider E, the space of bounded functions decomposing into a sum of pure transports and a “quasi compactly” supported part. We make a detailed analysis on the nonlinear interactions on E which leads us to make a restriction on the set of resonant phases.We finally give a convergence result which justifies the use of “quasi compactly” supported profiles.     

1-Soliton solution and conservation laws for nonlinear wave equation in semiconductors

This paper obtains the 1-soliton solution of a nonlinear wave equation that arises in the study of semiconductors. The conserved quantities are also calculated from this equation. Furthermore, additional non-trivial conserved quantities are computed using the invariance and multiplier approach based on the well known result that the Euler-Lagrange operator annihilates the total divergence.     

Decay estimation in nonlinear hyperbolic system of conservation laws

In this paper we study some decay estimates in nonlinear hyperbolic system of conservation laws. This research is not only interesting in itself but also crucial in studying the large time behavior problem. By introducing a proper Glimm functional, we obtain some useful decay estimates which are proved helpful in obtaining decay rates of the admissible solutions to nonlinear hyperbolic conservation laws as t→∞.     

On the reservoir technique convergence for nonlinear hyperbolic conservation laws - Part I

This paper is devoted to the analysis of flux schemes coupled with the reservoir technique for approximating hyperbolic equations and linear hyperbolic systems of conservation laws [F. Alouges, F. De Vuyst, G. Le Coq, E. Lorin, The reservoir scheme for systems of conservation laws, in: Finite Volumes for Complex Applications, III, Porquerolles, 2002, Lab. Anal. Topol. Probab. CNRS, Marseille, 2002, pp. 247-254 (electronic); F. Alouges, F. De Vuyst, G. Le Coq, E. Lorin, Un procédé de réduction de la diffusion numérique des schémas à différence de flux d'ordre un pour les systèmes hyperboliques non linéaires, C. R. Math. Acad. Sci. Paris 335 (7) (2002) 627-632; F. Alouges, F. De Vuyst, G. Le Coq, E. Lorin, The reservoir technique: A way to make Godunov-type schemes zero or very low diffusive. Application to Colella-Glaz, Eur. J. Mech. B Fluids 27 (6) (2008)]. We prove the long time convergence of the reservoir technique and its TVD property for some specific but still general configurations. Proofs are based on a precise study of the treatment by the reservoir technique of shock and rarefaction waves.     

On stability of linear and weakly nonlinear switched systems with time delay

This paper studies time-delayed switched systems that include both stable and unstable modes. By using multiple Lyapunov-functions technique and a dwell-time approach, several criteria on exponential stability for both linear and nonlinear systems are established. It is shown that by suitably controlling the switching between the stable and unstable modes, exponential stabilization of the switched system can be achieved. Some examples and numerical simulations are provided to illustrate our results.     

Roe solver with entropy corrector for uncertain hyperbolic systems

This paper deals with intrusive Galerkin projection methods with a Roe-type solver for treating uncertain hyperbolic systems using a finite volume discretization in physical space and a piecewise continuous representation at the stochastic level. The aim of this paper is to design a cost-effective adaptation of the deterministic Dubois and Mehlman corrector to avoid entropy-violating shocks in the presence of sonic points. The adaptation relies on an estimate of the eigenvalues and eigenvectors of the Galerkin Jacobian matrix of the deterministic system of the stochastic modes of the solution and on a correspondence between these approximate eigenvalues and eigenvectors for the intermediate states considered at the interface. We derive some indicators that can be used to decide where a correction is needed, thereby reducing the computational costs considerably. The effectiveness of the proposed corrector is assessed on the Burgers and Euler equations including sonic points.     

An optimal error estimate for upwind Finite Volume methods for nonlinear hyperbolic conservation laws

The purpose of this paper is to show that the cell-centered upwind Finite Volume scheme applied to general hyperbolic systems of m conservation laws approximates smooth solutions to the continuous problem at order one in space and time. As it is now well understood, there is a lack of consistency for order one upwind Finite Volume schemes: the truncation error does not tend to zero as the time step and the grid size tend to zero. Here, following our previous papers on scalar equations, we construct a corrector that allows us to prove the expected error estimate for nonlinear systems of equations in one dimension.     

Nonlinear waves in optical media

In nonlinear optical systems quasi-monochromatic waves satisfy the classical, constant dispersion and dispersion managed nonlocal, nonlinear Schrödinger equations, both of which exhibit localized pulse solutions. Current research has shown that mode-locked lasers are also described by dispersion managed equations. In spatial systems recent developments have attracted considerable interest in 2D photonic lattices. A computational method is introduced to find these and other localized waves in nonlinear optical media with vanishing and non-vanishing boundary conditions.     

A mixed bvp for flow in strongly anisotropic media

Initial-boundary value problems for a class of linear parabolic equations are considered. The anisotropy of the medium is characterised by a small parameter. The solution structure is analysed by singular perturbation methods which include the construction of outer solutions and boundary and initial layer terms. The analysis is justified by convergence results     

On the solution of a mixed nonlinear integral equation

In this paper, we consider a mixed nonlinear integral equation of the second kind in position and time. The existence of a unique solution of this equation is discussed and proved. A numerical method is used to obtain a system of Harmmerstein integral equations of the second kind in position. Then the modified Toeplitz matrix method, as a numerical method, is used to obtain a nonlinear algebraic system. Many important theorems related to the existence and uniqueness solution to the produced nonlinear algebraic system are derived. The rate of convergence of the total error is discussed. Finally, numerical examples when the kernel of position takes a logarithmic and Carleman forms, are presented and the error estimate, in each case, is calculated.     

Global weakly discontinuous solutions for hyperbolic conservation laws in the presence of a boundary

This work is a continuation of our previous work, in the present paper we study the mixed initial-boundary value problem for general n×n quasilinear hyperbolic systems of conservation laws with non-linear boundary conditions in the half space . Under the assumption that each characteristic with positive velocity is linearly degenerate, we prove the existence and uniqueness of global weakly discontinuous solution u=u(t,x) with small amplitude, and this solution possesses a global structure similar to that of the self-similar solution of the corresponding Riemann problem. Some applications to quasilinear hyperbolic systems of conservation laws arising in physics and other disciplines, particularly to the system describing the motion of the relativistic string in Minkowski space R1+n, are also given.     

Number and stability of nonlinear parametric oscillations in a system of weakly coupled oscillators with pendulum-type nonlinearity

The range of parameters for which the nonlinear system under study has three stable cycles with certain symmetry properties is determined. Translated fromMatematicheskie Zametki, Vol. 65, No. 3, pp. 369–376, March, 1999.     

On a new class of nonlinear set covering problems arising in wireless network design

This is a summary of the author’s PhD thesis, supervised by Edoardo Amaldi and defended on 28 April 2006 at Politecnico di Milano, Dipartimento di Matematica. The thesis is written in English and is available from the author upon request. The thesis investigates a class of nonlinear set covering variants arising from the problem of designing single-frequency Wireless Local Area Networks (WLANs) with maximum efficiency. In the first part of the thesis a basic hyperbolic formulation of the problem is considered. After a complexity and approximability study, the problem is tackled by linearization techniques, and by Lagrangean and Dantzig–Wolfe decompositions. The second part of the thesis focuses on variants accounting for various relevant features of the WLAN application. A Branch-and-Price algorithm is presented, and extensions to the multiple-frequency WLAN design problem are considered.      

Combination of mixed finite element and Dirichlet-to-Neumann methods in nonlinear plane elasticity

We study the solvability and Galerkin approximation of an exterior hyperelastic interface problem arising in plane elasticity. The weak formulation is obtained from an appropriate combination of a mixed finite element approach with a Dirichlet-to-Neumann method. The derivation of our results is based on some tools from nonlinear functional analysis and the Babuska-Brezzi theory for variational problems with constraints.     

Integrability aspects and soliton solutions for an inhomogeneous nonlinear system with symbolic computation

Under investigation in this paper is an inhomogeneous nonlinear system, which describes the marginally unstable baroclinic wave packets in geophysical fluids and ultra-short pulses in nonlinear optics under inhomogeneous media. Through symbolic computation, the Painlevé integrable condition, Lax pair and conservation laws are derived for this system. Furthermore, by virtue of the Darboux transformation, the explicit multi-soliton solutions are generated. Figures are plotted to reveal the following dynamic features of the solitons: (1) Parallel propagation of solitons: separation distance of the two parallel solitons depends on the value of |Im(λ₁)|-|Im(λ₂)| (where λ₁ and λ₂ are the spectrum parameters); (2) Periodic propagation of bound solitons: periodic bound solitons taking on contrary trends, and mutual attractions and repulsions of two bright bound solitons; (3) Elastic interactions of two one-peak bright solitons and of two one-peak dark solitons.