Vol. 12, No. 1, 2017

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ISSN: 2157-5452 (e-only)
ISSN: 1559-3940 (print)
A central-upwind geometry-preserving method for hyperbolic conservation laws on the sphere

Abdelaziz Beljadid and Philippe G. LeFloch

Vol. 12 (2017), No. 1, 81–107
Abstract

We introduce a second-order, central-upwind finite volume method for the discretization of nonlinear hyperbolic conservation laws on the two-dimensional sphere. The semidiscrete version of the proposed method is based on a technique of local propagation speeds, and the method is free of any Riemann solver. The main advantages of our scheme are its high resolution of discontinuous solutions, its low numerical dissipation, and its simplicity of implementation. We do not use any splitting approach, which is often applied to upwind schemes in order to simplify the resolution of Riemann problems. The semidiscrete form of our scheme is strongly built upon the analytical properties of nonlinear conservation laws and the geometry of the sphere. The curved geometry is treated here in an analytical way so that the semidiscrete form of the proposed scheme is consistent with a geometric compatibility property. Furthermore, the time evolution is carried out by using a total-variation diminishing Runge–Kutta method. A rich family of (discontinuous) stationary solutions is available for the conservation laws under consideration when the flux is nonlinear and foliated (in a suitable sense). We present a series of numerical tests, encompassing various nontrivial steady state solutions and therefore providing a good validation of the accuracy and efficiency of the proposed central-upwind finite volume scheme. Our numerical tests confirm that the scheme is stable and succeeds in accurately capturing discontinuous steady state solutions to conservation laws posed on the sphere.

Keywords
hyperbolic conservation law, shock wave, geometry-compatible flux, central-upwind scheme
Mathematical Subject Classification 2010
Primary: 35L65, 65M08
Secondary: 76L05
Milestones
Received: 28 March 2016
Revised: 31 December 2016
Accepted: 29 January 2017
Published: 8 May 2017
Authors
Abdelaziz Beljadid
Department of Civil and Environmental Engineering
Massachusetts Institute of Technology
77 Massachusetts Avenue
Cambridge, MA 02139
United States
Philippe G. LeFloch
Laboratoire Jacques-Louis Lions & Centre National de la Recherche Scientifique
Université Pierre et Marie Curie (Paris 6)
4 Place Jussieu
75258 Paris
France