A cost-effective nonlinear extremum-preserving finite volume scheme for highly anisotropic diffusion on Cartesian grids, with application to radiation belt dynamics

TL;DR

This paper introduces a new nonlinear finite volume scheme for highly anisotropic diffusion equations that preserves the minimum-maximum principle, converges faster, and demonstrates high accuracy and efficiency in radiation belt simulations.

Contribution

The paper presents a novel nonlinear FV scheme with a larger stencil that ensures positivity, faster convergence, and applicability to Cartesian and quadrilateral meshes for anisotropic diffusion.

Findings

Second-order convergence demonstrated in numerical tests.

Scheme shows improved computational efficiency over existing methods.

Effective in simulating anisotropic diffusion in radiation belt models.

Abstract

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.