CONSTRUCTION AND CONVERGENCE STUDY OF SCHEMES PRESERVING THE ELLIPTIC LOCAL MAXIMUM PRINCIPLE

Jerome Droniou (1) , Christophe Le Potier (2)

1 I3M - Institut de Mathématiques et de Modélisation de Montpellier
2 SFME - Service Fluide numériques, Modélisation et Etudes

We present a method to approximate (in any space dimension) diffusion equations with schemes having a specific structure; this structure ensures that the discrete local maximum and minimum principles are respected, and that no spurious oscillations appear in the solutions. When applied in a transient setting on models of concentration equations, it guaranties in particular that the approximate solutions stay between the physical bounds. We make a theoretical study of the constructed schemes, proving under a coercivity assumption that their solutions converge to the solution of the PDE. Several numerical results are also provided; they help us understand how the parameters of the method should be chosen. These results also show the practical efficiency of the method, even when applied to complex models.

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]

Jerome Droniou, Christophe Le Potier. CONSTRUCTION AND CONVERGENCE STUDY OF SCHEMES PRESERVING THE ELLIPTIC LOCAL MAXIMUM PRINCIPLE. SIAM Journal on Numerical Analysis, 2011, 49 (2), pp.459-490. ⟨10.1137/090770849⟩. ⟨hal-00808694⟩