Communications in Applied Mathematics and Computational Science

A central-upwind geometry-preserving method for hyperbolic conservation laws on the sphere

Abdelaziz Beljadid and Philippe LeFloch

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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.

Article information

Commun. Appl. Math. Comput. Sci., Volume 12, Number 1 (2017), 81-107.

Received: 28 March 2016
Revised: 31 December 2016
Accepted: 29 January 2017
First available in Project Euclid: 19 October 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 35L65: Conservation laws 65M08: Finite volume methods
Secondary: 76L05: Shock waves and blast waves [See also 35L67]

hyperbolic conservation law shock wave geometry-compatible flux central-upwind scheme


Beljadid, Abdelaziz; LeFloch, Philippe. A central-upwind geometry-preserving method for hyperbolic conservation laws on the sphere. Commun. Appl. Math. Comput. Sci. 12 (2017), no. 1, 81--107. doi:10.2140/camcos.2017.12.81.

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