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首页> 外文期刊>Journal of Computational Physics >Hyperbolic conservation laws on the sphere. A geometry-compatible finite volume scheme
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Hyperbolic conservation laws on the sphere. A geometry-compatible finite volume scheme

机译：球面上的双曲守恒定律。几何兼容的有限体积方案

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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.
机译：我们考虑与二维球面上标量非线性双曲守恒律相关的初值问题的熵解。我们提出一种有限体积方案，该方案依赖于由经度和纬度线段组成的网状网格。网格的结构允许自然几何相容性条件的离散形式，这是由Ben-Artzi和LeFloch建立的适度性理论提出的。我们在这里研究了几类通量矢量，它们定义了所考虑的守恒定律。它们基于在欧几里得三维空间中规定合适的矢量场，然后将其适当地投影在球体的切平面上的基础;即使当周围空间中的通量矢量恒定时，相应的通量矢量也是球上的非平凡矢量场。特别是，我们在这里构造“赤道周期解”，类似于一维守恒律的一维周期解以及各种各样的固定（稳态）解。我们还构造“受限解决方案”，这是随时间变化的解决方案，在球体的任意指定子域中受支持。最后，给出了代表性的数值示例和测试案例。

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