Abstract
A genuinely two-dimensional discretization of general drift-diffusion (including incompressible 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 and Gummel's in the sense that they contain modified Bessel functions which allow one 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.
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