Hyperbolic Conservation Laws on Manifolds. Total Variation Estimates and the Finite Volume Method
Hyperbolic Conservation Laws on Manifolds. Total Variation Estimates and the Finite Volume Method
复制标题
流形上的双曲守恒定律。
DOI：
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
P. LeFloch
中科院分区：
文献类型：
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作者：
M. Ben;P. LeFloch
本文研究了黎曼流形上双曲型守恒律方程熵解的一些性质。首先，我们推广的总变差递减（TVD）性质的流形，通过导出条件的流量的守恒律和一个给定的向量场，确保总变差的解决方案沿着积分曲线的向量场是不增加的时间。我们的结果是下一个专门的重要情况下的2球流，并讨论了通量的例子。其次，我们建立了基于数值流函数满足单调性的有限体积方法的收敛性。我们的证明需要详细的熵耗散的估计，并扩展到一般流形的早期证明由科伯恩，Coquel，和LeFloch在欧几里得的情况下。
This paper investigates some properties of entropy solutions of hyperbolic conservation laws on a Riemannian manifold. First, we generalize the Total Variation Diminishing (TVD) property to manifolds, by deriving conditions on the flux of the conservation law and a given vector field ensuring that the total variation of the solution along the integral curves of the vector field is non-increasing in time. Our results are next specialized to the important case of a flow on the 2-sphere, and examples of flux are discussed. Second, we establish the convergence of the finite volume methods based on numerical flux-functions satisfying monotonicity properties. Our proof requires detailed estimates on the entropy dissipation, and extends to general manifolds an earlier proof by Cockburn, Coquel, and LeFloch in the Euclidian case.