A geometry-preserving finite volume method for conservation laws on curved geometries

Abdelaziz Beljadid (1, 2) , Philippe G. LeFloch (1) , Abdolmajid Mohammadian (2)

1 LJLL - Laboratoire Jacques-Louis Lions
2 University of Ottawa - Department of Civil Engineering

We consider nonlinear hyperbolic equations posed on curved geometries and investigate a geometry-preserving, second-order accurate, finite volume method. For definiteness, we study the so-called class of ''geometric Burgers equations'' posed on the sphere and defined from a prescribed potential function. Despite its apparent simplicity, this model exhibits very complex wave phenomena that are not observed in absence of geometrical effects. Our method is based on second-order finite volume approximations and generalized Riemann solvers. Our main contribution is a rigorous investigation of the properties of discontinuous solutions. In particular, we demonstrate the contraction property, the time-variation monotonicity property, and the entropy monotonicity property (in all norms). We also study the late-time asymptotic behavior of solutions, which is found to depend on the properties of the flux vector and results from the nonlinear interactions between hyperbolic waves and the underlying geometry.

Domaines: Equations aux dérivées partielles [math.AP]

Abdelaziz Beljadid, Philippe G. LeFloch, Abdolmajid Mohammadian. A geometry-preserving finite volume method for conservation laws on curved geometries. 2013. ⟨hal-00922214⟩