Convergence of a positive nonlinear DDFV scheme for degenerate parabolic equations

El Houssaine Quenjel (1, 2, 3) , Mazen Saad (1, 2) , Mustapha Ghilani (4, 3) , Marianne Bessemoulin-Chatard (2, 5)

1 ECN - École Centrale de Nantes
2 LMJL - Laboratoire de Mathématiques Jean Leray
3 UMI - جامعة مولاي إسماعيل = Université Moulay Ismaïl
4 EMMACS
5 UN UFR ST - Université de Nantes - UFR des Sciences et des Techniques

In this work, we carry out the convergence analysis of a positive DDFV method for approximating solutions of degenerate parabolic equations. The basic idea rests upon different approximations of the fluxes on the same interface of the control volume. Precisely, the approximated flux is split into two terms corresponding to the primal and dual normal components. Then the first term is discretized using a centered scheme whereas the second one is approximated in a non evident way by an upstream scheme. The novelty of our approach is twofold: on the one hand we prove that the resulting scheme preserves the positivity and on the other hand we establish energy estimates. Some numerical tests are presented and they show that the scheme in question turns out to be robust and efficient. The accuracy is almost of second order on general meshes when the solution is smooth.

Mots clés: degenerate parabolic equations; Finite volume scheme; convergence; positive

Domaines: Analyse numérique [math.NA]

El Houssaine Quenjel, Mazen Saad, Mustapha Ghilani, Marianne Bessemoulin-Chatard. Convergence of a positive nonlinear DDFV scheme for degenerate parabolic equations. Calcolo, In press. ⟨hal-01757025v3⟩