TY - JOUR

T1 - A cost-effective nonlinear extremum-preserving finite volume scheme for highly anisotropic diffusion on Cartesian grids, with application to radiation belt dynamics

AU - Dahmen, Nour

AU - Droniou, Jérôme

AU - Rogier, François

N1 - Funding Information:
This research was supported by CNES – The French Space Agency, ONERA – The French Aerospace Lab, and the University of Monash .
Publisher Copyright:
© 2022 Elsevier Inc.

PY - 2022/8/15

Y1 - 2022/8/15

N2 - 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.

AB - 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.

KW - Anisotropic diffusion equation

KW - Discrete maximum principle

KW - Finite volume method

KW - Monotonicity

KW - Radiation belts

UR - http://www.scopus.com/inward/record.url?scp=85129964689&partnerID=8YFLogxK

U2 - 10.1016/j.jcp.2022.111258

DO - 10.1016/j.jcp.2022.111258

M3 - Article

AN - SCOPUS:85129964689

VL - 463

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

M1 - 111258

ER -