Dirichlet-to-Neumann mappings and finite-differences for anisotropic diffusion

•Contains an original method for discretizing two-dimensional parabolic PDE's based on Dirichlet-to-Neumann maps.•Explores the resulting schemes when the PDE's are standard heat or anisotropic diffusion equations on a Cartesian grid.•Derives numerical fluxes where external source terms are included into numerical fluxes.

A general methodology, which consists in deriving two-dimensional finite-difference schemes which involve numerical fluxes based on Dirichlet-to-Neumann maps (or Steklov-Poincaré operators), is first recalled. Then, it is applied to several types of diffusion equations, some being weakly anisotropic, endowed with an external source. Standard finite-difference discretizations are systematically recovered, showing that in absence of any other mechanism, like e.g. convection and/or damping (which bring Bessel and/or Mathieu functions inside that type of numerical fluxes), these well-known schemes achieve a satisfying multi-dimensional character.