Monotone and second order consistent schemes for the Pucci and Monge-Ampere equations

Joseph Frédéric Bonnans (1, 2) , Guillaume Bonnet (3) , Jean-Marie Mirebeau (4, 3)

1 X - École polytechnique
2 COMMANDS - Controle, Optimisation, modèles, Méthodes et Applications pour les Systèmes Dynamiques non linéaires
3 LMO - Laboratoire de Mathématiques d'Orsay
4 CNRS - Centre National de la Recherche Scientifique

We introduce a new strategy for the design of second-order accurate discretiza-tions of non-linear second order operators of Bellman type, which preserves degenerate ellipticity. The approach relies on Selling's formula, a tool from lattice geometry, and is applied to the Pucci and Monge-Ampere equations, discretized on a two dimensional cartesian grid. In the case of the Monge-Ampere equation, our work is related to both the stable formulation [FJ17] and the second order accurate scheme [BCM16]. Numerical experiments illustrate the robustness and the accuracy of the method.

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

Joseph Frédéric Bonnans, Guillaume Bonnet, Jean-Marie Mirebeau. Monotone and second order consistent schemes for the Pucci and Monge-Ampere equations. 2019. ⟨hal-02383521v1⟩