This paper studies a family of finite volume schemes for the hyperbolic scalar conservation law $u_t+{\nabla }_g·f(x,u)=0$ on a closed Riemannian manifold M. For an initial value in BV(M) we will show that these schemes converge with a $h^{\frac{1}{4}}$ convergence rate towards the entropy solution. When M is 1-dimensional the schemes are TVD and we will show that this improves the convergence rate to $h^{\frac{1}{2}}.$

This paper is devoted to the proof of almost global existence results for Klein-Gordon equations on compact revolution hypersurfaces with non-Hamiltonian nonlinearities, when the data are smooth, small and radial. The method combines normal forms with the fact that the eigenvalues associated to radial eigenfunctions of the Laplacian on such manifolds are simple and satisfy convenient asymptotic expansions.