  We construct a new nonlinear finite volume (FV) scheme for highly anisotropic
diffusion equations, that satisfies the discrete minimum-maximum principle. The
construction relies on the linearized scheme satisfying less restrictive
monotonicity conditions than those of an M-matrix, based on a weakly regular
matrix splitting and using the Cartesian structure of the mesh (extension to
quadrilateral meshes is also possible). The resulting scheme, obtained by
expressing fluxes as nonlinear combinations of linear fluxes, has a larger
stencil than other nonlinear positivity preserving or minimum-maximum principle
preserving schemes. Its larger "linearized" stencil, closer to the actual
complete stencil (that includes unknowns appearing in the convex combination
coefficients), enables a faster convergence of the Picard iterations used to
compute the solution of the scheme. Steady state dimensionless numerical tests
as well as simulations of the highly anisotropic diffusion in electron
radiation belts show a second order of convergence of the new scheme and
confirm its computational efficiency compared to usual nonlinear FV schemes.
