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We introduce a second-order, central-upwind finite volume method for thediscretization of nonlinear hyperbolic conservation laws posed on thetwo-dimensional sphere. The semi-discrete version of the proposed method isbased on a technique of local propagation speeds and it is free of any Riemannsolver. The main advantages of our scheme are the high resolution ofdiscontinuous solutions, its low numerical dissipation, and its simplicity forthe implementation. The proposed scheme does not use any splitting approach,which is applied in some cases to upwind schemes in order to simplify theresolution of Riemann problems. The semi-discrete form of the scheme isstrongly linked to the analytical properties of the nonlinear conservation lawand to the geometry of the sphere. The curved geometry is treated here in ananalytical way so that the semi-discrete form of the proposed scheme isconsistent with a geometric compatibility property. Furthermore, the timeevolution is carried out by using a total-variation-diminishing Runge-Kuttamethod. A rich family of (discontinuous) stationary solutions is available forthe problem under consideration when the flux is nonlinear and foliated (asidentified by the author in an earlier work). We present here a series ofnumerical examples, obtained by considering non-trivial steady state solutionsand this leads us to a good validation of the accuracy and efficiency of theproposed central-upwind finite volume method. Our numerical tests confirm thestability of the proposed scheme and clearly show its ability to captureaccurately discontinuous steady state solutions to nonlinear hyperbolicconservation laws posed on the sphere.

A central-upwind geometry-preserving method for hyperbolic conservation laws on the sphere