  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.
