In this paper an on-line algorithm for the Rectangle Packing Problem is presented. The method is designed to be able to accept or reject incoming boxes to maximize efficiency. We provide a wide computational analysis showing the behavior of the proposed algorithm as well as a comparison with existing off-line heuristics.

In this work we study a finite dynamical system for the description of the bifurcation pattern of the convection flow of a fluid between two parallel horizontal planes which undergoes a {\em horizontal} gradient of temperature ({\em horizontal} convection flow). Although in the two-dimensional case developed here,literature reports as well a long list of analytical and numerical solutions to this problem, the peculiar aim of this work makes it worthwhile. Actually we develop the route that Saltzman (1962) \cite{Sal62} and Lorenz (1963) \cite{Lor63} proposed for the {\em vertical} convection flow that started successfully the approach to finite dynamical systems. We obtain steady-to-steady and steady-to-periodic bifurcations in qualitative agreement with already published results. At first we adopt the non-dimensional scheme used by Saltzman and Lorenz; as it admits also physically meaningless solutions, we introduce a different set of reference quantities so overcoming this drawback.

The existence of global smooth solutions to the multi-dimensional hydrodynamic model for plasmas of electrons and positively charged ions is shown under the assumption that the initial densities are close to a constant. The model consists of the conservation laws for the particle densities and the current densities, coupled to the Poisson equation for the electrostatic potential. Furthermore, it is proved that the particle densities converge exponentially fast to the (constant) steady state. The proof uses a higher-order energy method inspired from extended thermodynamics.

We derive an asymptotic equation that describes the propagation of weakly nonlinear surface waves on a tangential discontinuity in incompressible magnetohydrodynamics. The equation is similar to, but simpler than, previously derived asymptotic equations for weakly nonlinear Rayleigh waves in elasticity. The most interesting feature of the surface waves is that their nonlinear self-interaction is nonlocal.

In refined network analysis, a compact network model is combined with distributed models for semiconductor devices in a multiphysics approach. For linear RLC networks containing diodes as distributed devices, we construct a mathematical model that combines the differential-algebraic network equations of the circuit with elliptic boundary value problems modeling the diodes. For this mixed initial-boundary value problem of partial differential-algebraic equations a first existence result is given.

The global existence of smooth solutions of the Cauchy problem for the $N$-dimensional Euler-Poisson model for semiconductors is established, under the assumption that the initial data is a perturbation of a stationary solution of the drift-diffusion equations with zero electron velocity, which is proved to be unique. The resulting evolutionary solutions converge asymptotically in time to the unperturbed state. The singular relaxation limit is also discussed.

We perform a multiple time scale, single space scale analysis of a compressible fluid in a time-dependent domain, when the time variations of the boundary are small with respect to the acoustic velocity. We introduce an average operator with respect to the fast time. The averaged leading order variables satisfy modified incompressible equations, which are coupled to linear acoustic equations with respect to the fast time. We discuss possible initial-boundary data for the asymptotic equations inherited from the initial-boundary data for the compressible equations.

We consider the Cauchy problem for a general one dimensional $n\times n$ hyperbolic symmetrizable system of balance laws. It is well known that, in many physical examples, for instance for the isentropic Euler system with damping, the dissipation due to the source term may prevent the shock formation, at least for smooth and small initial data. Our main goal is to find a set of general and realistic sufficient conditions to guarantee the global existence of smooth solutions, and possibly to investigate their asymptotic behavior. For systems which are entropy dissipative, a quite natural generalization of the Kawashima condition for hyperbolic-parabolic systems can be given. In this paper, we first propose a general framework to set this kind of problems, by using the so-called entropy variables. Therefore, we pass to prove some general statements about the global existence of smooth solutions, under different sets of conditions. In particular, the present approach is suitable for dealing with most of the physical examples of systems with a relaxation extension. Our main tools will be some refined energy estimates and the use of a suitable version of the Kawashima condition.

We study the deterministic counterpart of a backward-forward stochastic differential utility, which has recently been characterized as the solution to the Cauchy problem related to a PDE of degenerate parabolic type in two spatial variables, with a rank-1 diffusion and a conservative first order term. We first establish a local existence result for strong solutions and a continuation principle, and we produce a counterexample showing that, in general, strong solutions fail to be globally.

We introduce a differential model to study damage accumulation processes in presence of chemical reactions. The influence of micro-structure leads to a nonlinear parabolic system characterized by the presence of a characteristic length. Here, we first present an analytical description of the qualitative behavior of solutions which blow-up in finite time. Numerical simulations are given to describe the shape of solutions near the rupture time and the influence of the chemical reagents. Like in the non reactive model, the failure of the material occurs in a region of positive measure, due to the diffusive effects of the micro-structure, although some localization phenomena are observed. Moreover, if we increase the chemical concentration beyond a given threshold, which depends on the specific conditions of the material, we observe a strong acceleration in the damage process.

This paper deals with the development of a scientific computing environment for differential field simulation. We mean a modelling and simulation environment based on partial differential equations and their numerical solution as powerful and widely used technique for mathematical and computational investigation of application problems. We have been developing grid generation algorithms, numerical solvers of PDE systems, along with advanced visualization techniques, to numerically compute and evaluate field variables by exploiting user-friendly interaction. In this paper, we model the complete cycle of the visual computational simulation as reference framework and we illustrate advances in the environment development. We describe a few computational components by focusing on two fundamental substeps often conscurring to simulation processes, the image segmentation and grid generation. We introduce differential equation systems, developed combination of computational methods and recent algorithmic advances. A few application results are detailed, and shown by figures, for segmentation test problems.

The detailed analysis of a 1D-model for fluid flows in porous media with piecewise constant permeability clearly shows that variable permeability may lead to the phenomenon of oil trapping.

We consider the problem of writing Glimm type interaction estimates for the hyperbolic system \begin{equation}\label{E:abs0} u_t + A(u) u_x = 0. \end{equation} %only assuming that $A(u)$ is strictly hyperbolic. The aim of these estimates is to prove that there is Glimm-type functional $Q(u)$ such that \begin{equation}\label{E:abs1} \TV(u) + C_1 Q(u) \ \text{is lower semicontinuous w.r.t.} \ L^1-\text{norm}, \end{equation} with $C_1$ sufficiently large, and $u$ with small BV norm. In the first part we analyze the more general case of quasilinear hyperbolic systems. We show that in general this result is not true if the system is not in conservation form: there are Riemann solvers, identified by selecting an entropic conditions on the jumps, which do not satisfy the Glimm interaction estimate \eqref{E:abs1}. Next we consider hyperbolic systems in conservation form, i.e. $A(u) = Df(u)$. In this case, there is only one entropic Riemann solver, and we prove that this particular Riemann solver satisfies \eqref{E:abs1} for a particular functional $Q$, which we construct explicitly. The main novelty here is that we suppose only the Jacobian matrix $Df(u)$ strictly hyperbolic, without any assumption on the number of inflection points of $f$. These results are achieved by an analysis of the growth of $\TV(u)$ when nonlinear waves of \eqref{E:abs0} interact, and the introduction of a Glimm type functional $Q$, similar but not equivalent to Liu's interaction functional \cite{liu:admis}.

As strictly hyperbolic system of conservation laws of the form $$ u_{t}+f(u)_x =0 , \quad u(0,x)=\bar u (x)$$ is considered, where $ u \in\bbfR^N$, $f:\bbfR^N \rightarrow\bbfR^N$ is smooth, especially from a numerical point of view, that means, a semidiscrete upwind scheme of this equation is investigated. If we suppose that the initial data $\bar u (x) $ of this problem have small total variation the author proves that the solution of the upwind scheme $$ {\partial u(t,x) \over \partial t} + { ( f(u(t,x))-f(u(t,x-\varepsilon))) \over \varepsilon} =0 $$ has uniformly bounded variation (BV) norm independent on $t$ and $\varepsilon$. Moreover the Lipschitz-continuous dependence of the solution of the upwind scheme $u^{\varepsilon}(t)$ on the initial data is proved. This solution $u^{\varepsilon}(t)$ converges in $ L_1$ to a weak solution of the corresponding hyperbolic system as $ \varepsilon \rightarrow 0$. This weak solution coincides with the trajectory of a Riemann semigroup which is uniquely determined by the extension of Liu's Riemann solver to general hyperbolic systems.

The paper concerns with a hyperbolic system of conservation laws in one space variable $$ u_t + f(u)_x = 0,\qquad u(0,x) = u_0(x), $$ where $ u \in \Bbb R^n$, $f:\Omega \subseteq \Bbb R^n \rightarrow \Bbb R^n.$ Let $ K_0 \subset \Omega $ be a compact and let $\delta_1 > 0 $ be sufficiently small such that $K_1 = \{ u \in \Bbb R^n: \text{dist}(u,K_0) \leq \delta_1\}\subset \Omega.$ \par Assuming that the Jacobian matrix $A = Df$ is uniformly strictly hyperbolic in $K_1, u_0(-\infty) \in K_0$ and that the total variation of $u_0$ is sufficiently small, then there exists a unique ``entropic" solution $u: [0,+\infty) \rightarrow BV(\Bbb R,\Bbb R^n).$

We derive and study Well-Balanced schemes for quasimonotone discrete kinetic models. By means of a rigorous localization procedure, we reformulate the collision terms as nonconservative products and solve the resulting Riemann problem whose solution is self-similar. The construction of an Asymptotic Preserving (AP) Godunov scheme is straightforward and various compactness properties are established within different scalings. At last, some computational results are supplied to show that this approach is realizable and efficient on concrete $2 \times 2$ models.

Two systems of hyperbolic equations, arising in the multiphase semiclassical limit of the linear Schr\"odinger equations, are investigated. One stems from a Wigner measure analysis and uses a closure by the Delta functions, whereas the other relies on the classical WKB expansion and uses the Heaviside functions for closure. The two resulting moment systems are weakly and non-strictly hyperbolic respectively. They provide two different Eulerian methods able to reproduce superimposed signals with a finite number of phases. Analytical properties of these moment systems are investigated and compared. Efficient numerical discretizations and test-cases with increasing difficulty are presented.