  We introduce a second-order, central-upwind finite volume method for the
discretization of nonlinear hyperbolic conservation laws posed on the
two-dimensional sphere. The semi-discrete version of the proposed method is
based on a technique of local propagation speeds and it is free of any Riemann
solver. The main advantages of our scheme are the high resolution of
discontinuous solutions, its low numerical dissipation, and its simplicity for
the implementation. The proposed scheme does not use any splitting approach,
which is applied in some cases to upwind schemes in order to simplify the
resolution of Riemann problems. The semi-discrete form of the scheme is
strongly linked to the analytical properties of the nonlinear conservation law
and to the geometry of the sphere. The curved geometry is treated here in an
analytical way so that the semi-discrete form of the proposed scheme is
consistent with a geometric compatibility property. Furthermore, the time
evolution is carried out by using a total-variation-diminishing Runge-Kutta
method. A rich family of (discontinuous) stationary solutions is available for
the problem under consideration when the flux is nonlinear and foliated (as
identified by the author in an earlier work). We present here a series of
numerical examples, obtained by considering non-trivial steady state solutions
and this leads us to a good validation of the accuracy and efficiency of the
proposed central-upwind finite volume method. Our numerical tests confirm the
stability of the proposed scheme and clearly show its ability to capture
accurately discontinuous steady state solutions to nonlinear hyperbolic
conservation laws posed on the sphere.
