We examine spatially explicit models described by reaction-diffusion partial differential equations for the study of predator-prey population dynamics. The numerical methods we propose are based on the coupling of a finite difference/element spatial discretization and a suitable partitioned Runge-Kutta scheme for the approximation in time. The RK scheme here implemented uses an implicit scheme for the stiff diffusive term and a partitioned RK symplectic scheme for the reaction term (IMSP scheme).

In a recent paper, we investigated the uniform convergence of Lagrange interpolation at the zeros of the orthogonal polynomials with respect to a Freud-type weight in the presence of constraints. We show that by a simple procedure it is always possible to transform the matrices of these zeros into matrices such that the corresponding Lagrange interpolating polynomial with respect to the given constraints well approximates a given function.

In order to take into account the territory in which the outputs are in the market and the time-depending firms' strategies, the discrete Cournot duopoly game (with adaptive expectations, modeled by Kopel) is generalized through a non autonomous reaction-diffusion binary system of PDEs, with self and cross diffusion terms. Linear and nonlinear asymptotic $L^2$-stability, via the Lapunov Direct Method and a nonautonomous energy functional, are investigated.

The starting transient of highly under-expanded supersonic jets is studied by means of very high resolution weighted essentially non oscillatory finite volume schemes, coupled with a positivity-preserving scheme in order to ensure positivity of pressure and density for high compression/expansion ratio. Numerical behaviour of the schemes is investigated in terms of grid resolution, formal accuracy and different approximated Riemann solvers. The transient flow field is also discussed.

We use a framework that takes into account the effects of deformation of both the solid and fluid in the solidification process, to study the solidification of a semi-infinite layer of fluid. It is shown that the time required for solidification, and the final location of the interface are significantly different form the predictions of the classical Stefan problem. A detailed numerical solution of the initial-boundary value problem is provided for a variety of values for non-dimensional parameters relevant for freezing water.

We develop a numerical analysis of the buoyancy driven natural convection of a fluid in a three dimensional shallow cavity (4.1.1) with a horizontal gradient of temperature along the larger dimension. The fluid is a liquid metal (Prandtl number equal to 0.015) while the Grashof number (Gr) varies in the range 100,000-300,000. The Navier-Stokes equations in vorticity-velocity formulation have been integrated by means of a linearized fully implicit scheme. The evaluation of fractal dimension of the attractors in the phase space has allowed the detection of the chaotic regime.

We deal with the emergence of the horizontal three-dimensional convection flow from an asymptotic mechanical equilibrium in a parallelepipedic box with rigid walls and a very small horizontal temperature gradient. The non-linear stability bound is associated with a variational problem. It is proved that this problem is equivalent to the eigenvalue problem governing the linear stability pf the asymptotic basic conduction state and so the two bounds, the linear one and the non-linear one, coincide.

A vast literature exists on the Benard flow, the vertical thermal convection flow, but almost no result is known on the horizontal counterpart. On account of the wide range of applications in geophysics, astrophysics, metereology, and material science; we think that the horizontal thermal convection flow deserves as much consideration as the Benard problem. The present study is the first step towards the description of the bifurcation pattern of the horizontal thermal convection flow.

We propose a time-advancing scheme for the discretization of the unsteady incompressible Navier-Stokes equations. At any time step, we are able to decouple velocity and pressure by solving some suitable elliptic problems. In particular, the problem related with the determination of the pressure does not require boundary conditions. The divergence free condition is imposed as a penalty term, according to an appropriate restatement of the original equations. Some experiments are carried out by approximating the space variables with the spectral Legendre collocation method.

A generalization of mesoscopic Lattice-Boltzmann models aimed at describing flows with solid/liquid phase transitions is presented. It exhibits lower computational costs with respect to the numerical schemes resulting from differential models, Moreover it is suitable to describe chaotic motions in the mushy zone.