A Godunov scheme is derived for two-dimensional scalar conservation laws without or with source terms following ideas originally proposed by Boukadida and LeRoux [Math. Comput., 63 (1994), pp. 541-553] in the context of a staggered Lax-Friedrichs scheme. In both situations, the numerical fluxes are obtained at each interface of a uniform Cartesian computational grid just by means of the "external waves" involved in the entropy solution of the elementary two-dimensional (2D) Riemann problems; in particular, all the wave-interaction phenomena are overlooked.

When numerically simulating a kinetic model of an n+nn+ semiconductor device, obtaining a constant macroscopic current at steady state is still a challenging task. Part of the difficulty comes from the multiscale, discontinuous nature of both p|n junctions, which create spikes in the electric field and enclose a channel where corresponding depletion layers glue together. The kinetic formalism furnishes a model holding inside the whole domain, but at the price of strongly varying parameters.

The numerical approximation of one-dimensional relativistic Dirac wave equations is considered within the recent framework consisting in deriving local scattering matrices at each interface of the uniform Cartesian computational grid. For a Courant number equal to unity, it is rigorously shown that such a discretization preserves exactly the (Formula presented.) norm despite being explicit in time. This construction is well-suited for particles for which the reference velocity is of the order of (Formula presented.), the speed of light.

Tidal interactions have a significant influence on the late dynamics of compact binary systems, which constitute the prime targets of the upcoming network of gravitational-wave detectors. We refine the theoretical description of tidal interactions (hitherto known only to the second post-Newtonian level) by extending our recently developed analytic self-force formalism, for extreme-mass-ratio binary systems, to the computation of several tidal invariants.

We review recent advances on the mesoscopic modeling of water-like fluids, based on the lattice Boltzmann (LB) methodology. The main idea is to enrich the basic LB (hydro)-dynamics with angular degrees of freedom responding to suitable directional potentials between water-like molecules. The model is shown to reproduce some microscopic features of liquid water, such as an average number of hydrogen bonds per molecules (HBs) between 3 and 4, as well as a qualitatively correct statistics of the hydrogen bond angle as a function of the temperature.

One of the promising frontiers of bioengineering is the controlled release of a therapeutic drug from a vehicle across the skin (transdermal drug delivery). In order to study the complete process, a two-phase mathematical model describing the dynamics of a substance between two coupled media of different properties and dimensions is presented. A system of partial differential equations describes the diffusion and the binding/unbinding processes in both layers. Additional flux continuity at the interface and clearance conditions into systemic circulation are imposed.