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.

We study the phase separation of a binary mixture in uniform shear flow in the framework of the continuum convection-diffusion equation based on a Ginzburg-Landau free energy. This equation is solved both numerically and in the context of large-N approximation. Our results show the existence of domains with two typical sizes, whose relative abundance changes in time. As a consequence log-time periodic oscillations are observed in the behavior of most thermodynamic observables.

We present a numerical study of the dynamics of a non-ideal fluid subject to a density-dependent pseudo-potential characterized by a hierarchy of nested attractive and repulsive interactions. It is shown that above a critical threshold of the interaction strength, the competition between stable and unstable regions results in a short-ranged disordered fluid pattern with sharp density contrasts. These disordered configurations contrast with phase-separation scenarios typically observed in binary fluids. The present results indicate that frustration can be modelled within the framework of a suitable one-body effective Boltzmann equation. The lattice implementation of such an effective Boltzmann equation may be seen as a preliminary step towards the development of complementary/alternative approaches to truly atomistic methods for the computational study of glassy dynamics.

Si dimostra come applicare il Principio di Massimo Ibrido e il Principio Necessario Ibrido ad un sistema di controllo che modellizza una macchina con marce.

The simplest equation for the evolution of a director field is given by its corresponding heat flow. More complicated versions arise in the theories of micromagnetism and liquid crystals. In 3D there exist finite energy solutions with point singularities (also called defects in case of liquid crystals). It the paper an example of a new nonuniqueness phenomenon is discussed: having initially an equilibrium situation with one point singularity, a solution is constructed for which the singularity is moved instantaneously to another point. This suggests that there exists a considerable degree of freedom to prescribe the evolution of point singularities.

The expression of a bound of the uniform norm of infinite lower triangular Toeplitz matrices with nonnegative entries is found. All the results are obtained by studying the behavior of the resolvent kernel and of the fundamental matrix of the recurrence relation which generates the sequence of the entries of the considered matrix.

We develop a numerical method to study the dynamics of a two-component atomic Fermi gas trapped inside a harmonic potential at temperature T well below the Fermi temperature TF. We examine the transition from the collisionless to the collisional regime down to T = 0.2 TF and find a good qualitative agreement with the experiments of B. DeMarco and D.S. Jin [Phys. Rev. Lett. 88, 040405 (2002)]. We demonstrate a twofold role of temperature on the collision rate and on the efficiency of collisions. In particular, we observe a hitherto unreported effect, namely, the transition to hydrodynamic behavior is shifted towards lower collision rates as temperature decreases.