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

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-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.

Diffusive limit of a two-dimensional well-balanced approximation to a kinetic model of chemotaxis

“…The scheme (4.4) relaxes, as e ! 0, towards a discretization of drift-diffusion equation and shares some features with the one previously studied in [2], especially for the presence of modified Bessel functions in its numerical fluxes. However, they aren't strictly identical, despite both emerge from similar Fourier-Bessel expressions of stationary drift-diffusion solutions.…”

“…The restriction to a ''four-velocity model'' like (1.1) is harder to circumvent: indeed, in order to take full advantage of the S-matrix (3.3), one needs to have grid points matching exactly incoming/outgoing states; here we chose Delaunay circles corresponding to square grid cells (see Fig. 2), like in [2,22,23]. In return, the process produces a remarkable scheme, especially in its IMEX version (4.2) which can be proved to be both well-balanced and asymptotic-preserving, see Theorem 4.1.…”

“…Recently, efforts were drawn onto the derivation of ''truly 2D well-balanced'' discretizations for various linear (or weakly nonlinear) equations, see for instance [2,28], which go beyond what was done for ð1 þ 1Þ-dimensional models in [20, Part II] and following papers [7,21,26,27]. The derivation of two-dimensional numerical schemes for which it is possible to prove that exact (or truncated, see below) steady-states are left invariant, relies on two main ingredients:…”

A Two-Dimensional “Flea on the Elephant” Phenomenon and its Numerical Visualization

“…Such regularity in a square domain requires smooth boundary data to be supplemented by compatibility conditions (A.2) at each corner, in order to apply Theorem A.2. This is a situation closely related to "well-balanced methods" in 1D, see also [3,16,19] for 2D considerations. In particular, the modified Bessel functions contained in (4.11), which numerically allow not to split between the Laplacian and the zero-order term, moreover can fit the sharp layers appearing in the solution in case the potential becomes stiff (like "exponentialfit" methods in 1D).…”

Numerical scheme for kinetic transport equation with internal state *

“…Truly two-dimensional well-balanced schemes are not much developed yet. We mention the recent paper [1], and [9] in the particular case of radiative transfer equation. However, the case at hand can be seen as a one dimension problem by introducing a new variable.…”