Monotone corrections for generic cell-centered Finite Volume approximations of anisotropic diffusion equations

Clément Cancès (1) , Mathieu Cathala (2) , Christophe Le Potier (3)

1 LJLL - Laboratoire Jacques-Louis Lions
2 I3M - Institut de Mathématiques et de Modélisation de Montpellier
3 CEA-DES (ex-DEN) - CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire)

We present a nonlinear technique to correct a general Finite Volume scheme for anisotropic diffusion problems, which provides a discrete maximum principle. We point out general properties satisfied by many Finite Volume schemes and prove the proposed corrections also preserve these properties. We then study two specific corrections proving, under numerical assumptions, that the corresponding solutions converge to the continuous one as the size of the mesh tends to 0. Finally we present numerical results showing these corrections suppress local minima produced by the initial Finite Volume scheme.

Mots clés: Finite Volume scheme; Diffusion equation; Anisotropy; Maximum principle; Nonlinear corrections; Convergence

Domaines: Analyse numérique [math.NA]

Clément Cancès, Mathieu Cathala, Christophe Le Potier. Monotone corrections for generic cell-centered Finite Volume approximations of anisotropic diffusion equations. 2013. ⟨hal-00643838v2⟩