This paper is devoted to the study of a turbulentcirculation model. Equations are derived from the “Navier-Stokes turbulentkinetic energy” system. Some simplifications are performed but attentionis focused on non linearities linked to turbulent eddy viscosity  $\nu _{t}$ . The mixing length $\ell $ acts as a parameter which controls theturbulent part in $\nu _{t}$ . The main theoretical results that we haveobtained concern the uniqueness of the solution for bounded eddy viscositiesand small values of $\ell $ and its asymptotic decreasing as $\ell\rightarrow \infty $ in more general cases. Numerical experimentsillustrate but also allow to extend these theoretical results: uniqueness isproved only for $\ell $ small enough while regular solutions are numericallyobtained for any values of $\ell $ . A convergence theorem is proved forturbulent kinetic energy: $k_{\ell }\rightarrow 0$ as $\ell \rightarrow\infty ,$ but for velocity $u_{\ell }$ we obtain only weaker results.Numerical results allow to conjecture that $k_{\ell }\rightarrow 0,$ $\nu_{t}\rightarrow \infty $ and $u_{\ell }\rightarrow 0$ as $\ell \rightarrow\infty .$ So we can conjecture that this classical turbulent model obtainedwith one degree of closure regularizes the solution.

We investigate sufficient and possiblynecessary conditions for the L 2 stability of the upwind first orderfinite volume scheme for Maxwell equations, with metallic andabsorbing boundary conditions. We yield a very general sufficient condition,valid for any finite volume partition in two and three spacedimensions. We show this condition is necessary for a class ofregular meshes in two space dimensions. However, numerical tests show it is not necessaryin three space dimensions even on regular meshes. Stability limits for time and space schemes with higher orders of accuracy are numericallyinvestigated.