Hyperbolic conservation laws on the sphere. A geometry-compatible finite volume scheme

Matania Ben-Artzi (1) , Joseph Falcovitz (1) , Philippe G. LeFloch (2)

1 Institute of Mathematics - Hebrew University
2 LJLL - Laboratoire Jacques-Louis Lions

We consider entropy solutions to the initial value problem associated with scalar nonlinear hyperbolic conservation laws posed on the two-dimensional sphere. We propose a finite volume scheme which relies on a web-like mesh made of segments of longitude and latitude lines. The structure of the mesh allows for a discrete version of a natural geometric compatibility condition,which arose earlier in the well-posedness theory established by Ben-Artzi and LeFloch. We study here several classes of flux vectors which define the conservation law under consideration.They are based on prescribing a suitable vector field in the Euclidean three-dimensional space and then suitably projecting it on the sphere's tangent plane; even when the flux vector in the ambient space is constant, the corresponding flux vector is a non-trivial vector field on the sphere. In particular, we construct here "equatorial periodic solutions", analogous to one-dimensional periodic solutions to one-dimensional conservation laws, as well as a wide variety of stationary (steady state) solutions. We also construct "confined solutions", which are time-dependent solutions supported in an arbitrarily specified subdomain of the sphere. Finally,representative numerical examples and test-cases are presented.

Mots clés: <i>Hyperbolic conservation law; Sphere; Entropy solution; Finite volume scheme; Geometry-compatible flux.</i>

Domaines: Analyse numérique [cs.NA]

Matania Ben-Artzi, Joseph Falcovitz, Philippe G. LeFloch. Hyperbolic conservation laws on the sphere. A geometry-compatible finite volume scheme. 2009. ⟨hal-00387395⟩