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Pré-Publication, Document De Travail Année : 2016

A second-order geometry-preserving finite volume method for conservation laws on the sphere

Résumé

We consider nonlinear hyperbolic conservation laws posed on curved geometries —referred to as ``geometric Burgers equations" after Ben-Artzi and LeFloch— when the underlying geometry is the two-dimensional sphere and the flux vector field is determined from a potential function. Despite its apparent simplicity, this hyperbolic model exhibits complex wave phenomena that are not observed in absence of geometrical effects. We formulate a second-order accurate, finite volume method which is based on a latitude/longitude triangulation of the sphere and on a generalized Riemann solver and a direction splitting based on the sphere geometry. Importantly, this scheme is geometry-preserving in the sense that the discrete form of the scheme respects the divergence free condition for the conservation law on the sphere. A total variation diminishing Runge-Kutta method with an operator splitting approach is used for temporal integration. The quality of the numerical solutions is largely improved using the proposed piecewise linear reconstruction and the method performs well for discontinuous solutions with large amplitude and shocks in comparison with the existing schemes. With this method, we numerically investigate the properties of discontinuous solutions and numerically demonstrate the contraction, time-variation monotonicity, and entropy monotonicity properties. Next, we study the late-time asymptotic behavior of solutions, and discuss it in terms of the properties of the flux vector field. We thus provide a rigorous validation of the accuracy and efficiency of the proposed finite volume method in presence of nonlinear hyperbolic waves and a curved geometry. The method should be extendable to the shallow water model posed on the sphere.
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Dates et versions

hal-01423815 , version 1 (31-12-2016)

Identifiants

  • HAL Id : hal-01423815 , version 1

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Abdelaziz Beljadid, Philippe G Lefloch, Abdolmajid Mohammadian. A second-order geometry-preserving finite volume method for conservation laws on the sphere. 2016. ⟨hal-01423815⟩
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