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

43 pagesWe 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