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Finite Volume Methods and Adaptive Refinement for Tsunami Propagation and Inundation
, 2006
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Hyperbolic conservation laws on manifolds. Error estimate for finite volume schemes
 Acta Math. Sinica
, 2009
"... Following BenArtzi and LeFloch, we consider nonlinear hyperbolic conservation laws posed on a Riemannian manifold, and we establish an L 1error estimate for a class of finite volume schemes allowing for the approximation of entropy solutions to the initial value problem. The error in the L 1 norm ..."
Abstract

Cited by 4 (3 self)
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Following BenArtzi and LeFloch, we consider nonlinear hyperbolic conservation laws posed on a Riemannian manifold, and we establish an L 1error estimate for a class of finite volume schemes allowing for the approximation of entropy solutions to the initial value problem. The error in the L 1 norm is of order h 1/4 at most, where h represents the maximal diameter of elements in the family of geodesic triangulations. The proof relies on a suitable generalization of Cockburn, Coquel, and LeFloch’s theory which was originally developed in the Euclidian setting. We extent the arguments to curved manifolds, by taking into account the effects to the geometry and overcoming several new technical difficulties. 1 Introduction and
A CONVERGENCE RESULT FOR FINITE VOLUME SCHEMES ON 2DIMENSIONAL RIEMANNIAN MANIFOLDS
"... 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 2dimensional manifold we will show that these schemes converge with a h 1 4 convergence rate t ..."
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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 2dimensional manifold we will show that these schemes converge with a h 1 4 convergence rate towards the entropy solution. When M is 1dimensional 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 magnetogravity 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 nonlinear 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 2dimensional 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 − κϕt + g(x) ( ( ˜ f(u⊤κ) − ˜ f(u⊥κ))v(x), ∇gϕ)] (3)