For some models of relevance in the social sciences we review some examples in which system size plays an important role in the final outcome of the dynamics. We discuss the conditions under which changes of behavior can appear only when the number of agents in the model takes a finite value. Those changes of behavior can be related to the apparent phase transitions that appear in some physical models. We show examples in the Galam's model of opinion transmission and the Axelrod's model of culture formation stressing the role that the network of interactions has on the main results of both models. Finally, we present the phenomenon of system-size stochastic resonance by which a forcing signal (identified as an advertising agent) is optimally amplified by a population of the right (intermediate) size. Our work stresses the role that the system size has in the dynamics of social systems and the inappropriateness of taking the thermodynamic limit for these systems.

Concepts of the lattice Boltzmann method are discussed in detail for the one-dimensional kinetic model. Various techniques of constructing lattice Boltzmann models are discussed, and novel collision integrals are derived. Geometry of the kinetic space and the role of the thermodynamic projector is elucidated. 

In this paper, the three-dimensional lattice kinetic scheme is presented to simulate incompressible viscous thermal flows. As compared with the standard LBM, the present scheme has the following good features. It can save the computer memory since there is no need to store the density distributions. Like the conventional NS solvers, the implementation of boundary conditions is straightforward since the dependent variables are the macroscopic flow parameters. The easy implementation of boundary conditions is a good property for solving three-dimensional flow problems. The present scheme is validated by simulating the three-dimensional natural convection in an air-filled cubical enclosure, which is heated differentially at two vertical side walls. The obtained numerical results compare very well with available data in the literature.

In this work, we will present a system of numerical simulations for inertial confinement fusion, which consists of a series of one-dimensional, two-dimensional and three-dimensional codes. Our efforts have been made to develop 2D and 3D computer simulation codes, forming a 2D simulation capability so with the key physics issues in laser fusion can be separately simulated mainly by a LARED family containing six different 2D (and partially 3D) code series. The models and the characteristics of the main codes will be described, and some simulation results using the LARED family will be presented.

We describe and evaluate a numerical solution strategy for simulating surface acoustic waves (SAWs) through semiconductor devices with complex geometries. This multi-physics problem is of particular relevance to the design of SAW-based quantum electronic devices. The mathematical model consists of two coupled partial differential equations for the elastic wave propagation and the electric field, respectively, in anisotropic piezoelectric media. These equations are discretized by the finite element method in space and by a finite difference method in time. The latter method yields a convenient numerical decoupling of the governing equations. We describe how a computer implementation can utilize the decoupling and, via object-oriented programming techniques reuse independent codes for the Poisson equation and the linear time-dependent elasticity equation. First we apply the simulator to a simplified model problem for verifying the implementation, and thereafter we show that the methodology is capable of simulating a real-world case from nanotechnology, involving SAWs in a geometrically non-trivial device made of Gallium Arsenide.

A numerical algorithm for effective incorporation of parametric uncertainty into mathematical models is presented. The uncertain parameters are modeled as random variables, and the governing equations are treated as stochastic. The solutions, or quantities of interests, are expressed as convergent series of orthogonal polynomial expansions in terms of the input random parameters. A high-order stochastic collocation method is employed to solve the solution statistics, and more importantly, to reconstruct the polynomial expansion. While retaining the high accuracy by polynomial expansion, the resulting “pseudo-spectral” type algorithm is straightforward to implement as it requires only repetitive deterministic simulations. An estimate on error bounded is presented, along with numerical examples for problems with relatively complicated forms of governing equations. 

A high order multidomain spectral difference method has been developed for the three dimensional Navier-Stokes equations on unstructured hexahedral grids. The method is easy to implement since it involves one-dimensional operations only, and does not involve surface or volume integrals. Universal reconstructions are obtained by distributing solution and flux points in a geometrically similar manner in a unit cube. The concepts of the Riemann solver and high-order local representations are applied to achieve conservation and high order accuracy. In this paper, accuracy studies are performed to numerically verify the order of accuracy using flow problems with analytical solutions. High order of accuracy and spectral convergence are obtained for the propagation of an isotropic vortex and Couette flow. The capability of the method for both inviscid and viscous flow problems with curved boundaries is also demonstrated.

We consider the Thomas-Fermi-von Weizsacker energy functional, with the Wang-Teter correction, and present an efficient real space method for Orbital-Free Density Functional Theory. It is proved that the energy minimizer satisfies a second order quasilinear elliptic equation, even at the points where the electron density vanishes. This information is used to construct an efficient energy minimization method for the resulting constrained problem, based on the truncated Newton method for unconstrained optimization. The Wang-Teter kernel is analyzed, and its behavior in real space at short and far distances is determined. A second order accurate discretization of the energy is obtained using finite differences. The efficiency and accuracy of the method is illustrated with numerical simulations in an Aluminium FCC lattice.

On Barabási-Albert networks with z neighbours selected by each added site, the Ising model was seen to show a spontaneous magnetisation. This spontaneous magnetisation was found below a critical temperature which increases logarithmically with system size. On these networks the majority-vote model with noise is now studied through Monte Carlo simulations. However, in this model, the order-disorder phase transition of the order parameter is well defined in this system and this was not found to increase logarithmically with system size. We calculate the value of the critical noise parameter q_c for several values of connectivity z of the undirected Barabási-Albert network. The critical exponentes β/ν, γ/ν and 1/ν were also calculated for several values of z.