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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 For an initial value in BV() we will show that these schemes converge with a $h^{\frac{1}{4}}$ convergence rate towards the entropy solution. When is -dimensional the schemes are TVD and we will show that this improves the convergence rate to $h^{\frac{1}{2}}.$

The notion of singular limiting induced from continuum solutions (slic-solutions) is applied to the problem of cavitation in nonlinear elasticity, in order to re-assess an example of non-uniqueness of entropic weak solutions (with polyconvex energy) due to a forming cavity.