Hyperbolic conservation laws on the sphere. A geometry-compatible finite volume scheme

JOURNAL OF COMPUTATIONAL PHYSICS

卷:228

Article

Ben-Artzi, Matania²  Falcovitz, Joseph²  LeFloch, Philippe G.¹ 

[1] Univ Paris 06, CNRS, Lab Jacques Louis Lions, F-75252 Paris, France

[2] Hebrew Univ Jerusalem, Inst Math, IL-91904 Jerusalem, Israel

关键词: Hyperbolic conservation law;    Sphere;    Entropy solution;    Finite volume scheme;    Geometry-compatible flux;   

DOI  :  10.1016/j.jcp.2009.04.032

来源: Elsevier

【 摘 要 】

We consider entropy solutions to the initial value problem associated with scalar nonlinear hyperbolic conservation laws posed on the two-dimensional sphere. We propose a finite volume scheme which relies on a web-like mesh made of segments of longitude and latitude lines. The structure of the mesh allows for a discrete version of a natural geometric compatibility condition, which arose earlier in the well-posedness theory established by Ben-Artzi and LeFloch. We study here several classes of flux vectors which define the conservation law under consideration. They are based on prescribing a suitable vector field in the Euclidean three-dimensional space and then suitably projecting it on the sphere's tangent plane; even when the flux vector in the ambient space is constant, the corresponding flux vector is a non-trivial vector field on the sphere. In particular, we construct here equatorial periodic solutions, analogous to one-dimensional periodic solutions to one-dimensional conservation laws, as well as a wide variety of stationary (steady state) solutions. We also construct confined solutions, which are time-dependent solutions supported in an arbitrarily specified subdomain of the sphere. Finally, representative numerical examples and test cases are presented. (C) 2009 Elsevier Inc. All rights reserved.

【 授权许可】

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