Viscous Equations Treated with L-Splines and Steklov-Poincaré Operator in Two Dimensions

Well-balanced schemes, nowadays well-known for 1D hyperbolic equations with source terms and systems of balance laws, are extended to strictly parabolic equations, first in 1D, then in 2D on Cartesian computational grids. The construction heavily relies on a particular type of piecewise-smooth interpolation of discrete data known as -splines. In 1D, several types of widely-used discretizations are recovered as particular cases, like the El-Mistikawy-Werle scheme or Scharfetter- Gummel's. Moreover, a distinctive feature of our 2D scheme is that dimensional-splitting never occurs within its derivation, so that all the multi-dimensional interactions are kept at the discrete level. This leads to improved accuracy, as illustrated on several types of drift-diffusion equations.