Convergence order of upwind type schemes for transport equations with discontinuous coefficients

François Delarue (1) , Frédéric Lagoutière (2) , Nicolas Vauchelet (3)

1 LJAD - Laboratoire Jean Alexandre Dieudonné
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay
3 LJLL - Laboratoire Jacques-Louis Lions

An analysis of the error of the upwind scheme for transport equation with discontinuous coefficients is provided. We consider here a velocity field that is bounded and one-sided Lipschitz continuous. In this framework, solutions are defined in the sense of measures along the lines of Poupaud and Rascle's work. We study the convergence order of the upwind scheme in the Wasserstein distances. More precisely, we prove that in this setting the convergence order is 1/2. We also show the optimality of this result. In the appendix, we show that this result also applies to other " diffusive " " first order " schemes and to a forward semi-Lagrangian scheme.

Mots clés: upwind finite volume scheme; convergence order; forward semi-Lagrangian scheme; conservative transport equation; measure-valued solution; continuity equation

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

François Delarue, Frédéric Lagoutière, Nicolas Vauchelet. Convergence order of upwind type schemes for transport equations with discontinuous coefficients. Journal de Mathématiques Pures et Appliquées, 2017. ⟨hal-01273848⟩