Monotone and Consistent discretization of the Monge-Ampere operator

Jean-David Benamou (1) , Francis Collino (2) , Jean-Marie Mirebeau (3)

1 MOKAPLAN - Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales
2 CERFACS - Centre Européen de Recherche et de Formation Avancée en Calcul Scientifique
3 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

We introduce a novel discretization of the Monge-Ampere operator, simultaneously consistent and degenerate elliptic, hence accurate and robust in applications. These properties are achieved by exploiting the arithmetic structure of the discrete domain, assumed to be a two dimensional cartesian grid. The construction of our scheme is simple, but its analysis relies on original tools seldom encountered in numerical analysis, such as the geometry of two dimensional lattices, and an arithmetic structure called the Stern-Brocot tree. Numerical experiments illustrate the method's efficiency.

Mots clés: Stern-Brocot tree.; Stern-Brocot tree; Monge Ampere PDE; Monotone Finite differences scheme; Lattice Basis reduction

Domaines: Analyse numérique [math.NA]

Jean-David Benamou, Francis Collino, Jean-Marie Mirebeau. Monotone and Consistent discretization of the Monge-Ampere operator. Mathematics of Computation, 2016, 85 (302), pp.2743-2775. ⟨10.1090/mcom/3080⟩. ⟨hal-01067540⟩