A Truly Two-Dimensional Discretization of Drift-Diffusion Equations on Cartesian Grids

Roberta Bianchini (1) , Laurent Gosse (2)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
2 IAC - Istituto per le Applicazioni del Calcolo "Mauro Picone"

A genuinely two-dimensional discretization of general drift-diffusion (including in-compressible Navier-Stokes) equations is proposed. Its numerical fluxes are derived by computing the radial derivatives of "bubbles" which are deduced from available discrete data by exploiting the stationary Dirichlet-Green function of the convection-diffusion operator. These fluxes are reminiscent of Scharfetter-Gummel's in the sense that they contain modified Bessel functions which allow to pass smoothly from diffusive to drift-dominating regimes. For certain flows, monotonicity properties are established in the vanishing viscosity limit ("asymptotic monotony") along with second-order accuracy when the grid is refined. Practical benchmarks are displayed to assess the feasibility of the scheme, including the "western currents" with a Navier-Stokes-Coriolis model of ocean circulation.

Mots clés: Bubbles; Drift-Diffusion; Green-Dirichlet function; Navier-Stokes-Coriolis

Domaines: Mathématiques [math]

Roberta Bianchini, Laurent Gosse. A Truly Two-Dimensional Discretization of Drift-Diffusion Equations on Cartesian Grids. SIAM Journal on Numerical Analysis, 2018, 56 (5), pp.2845-2870. ⟨10.1137/17m1151353⟩. ⟨hal-02012706⟩