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Abstract. This paper studies a family of nite volume schemes for the hyperbolic scalar conservation law ut + ∇g · 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. 1. introduction Hyperbolic partial di erential equations on curved manifolds occur in many applications. These include shallow water models for the atmosphere or ocean [4], [13], [16], the propagation of sound waves on curved surfaces [21] and passive tracer advection in the atmosphere. Further examples are the propagation of magneto-gravity waves in the solar tachocline [20], [5], [10] and relativistic matter ows near compact objects like black holes [9], [14]. For the numerics of these problems nite di erence [9], nite volume [14], discontinuous Galerkin [12] and wave propagation methods [19] have been used. For convergence analysis of nite volume schemes, we will consider the following scalar model problem for non-linear hyperbolic conservation laws: (1) (2) ut + ∇g · ( ˜ f(u)v(x)) = 0 in M × R+ u(x, 0) = u0(x) on M. Here (M, g) is a 1- or 2-dimensional closed oriented Riemannian manifold, v is a smooth vector- eld on M and g is a xed Riemannian metric on M. By ∇g · we denote the divergence operator on M induced by g. The aim of this paper is to prove a convergence rate for nite volume schemes for this model problem. For this problem one has the notion of entropy solution, analogous to the Kruzkov de nition in Euclidean space. De nition 1. A function u ∈ L ∞ (M × R+) is called an entropy solution of (1),(2) if |u − κ|ϕt + g(x) ( ( ˜ f(u⊤κ) − ˜ f(u⊥κ))v(x), ∇gϕ)] (3)