Abstract: This paper studies a family of finite volume schemes for the hyperbolic scalar conservation law u_t + div f(x,u) = 0 on a closed Riemannian manifold. For an initial value in BV(M) and an at most 2-dimensional manifold we will show that these schemes converge with a h^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^1/2.