Consistency of finite volume approximations to nonlinear hyperbolic balance laws
Consistency of finite volume approximations to nonlinear hyperbolic balance laws
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DOI：
10.1090/mcom/3569
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发表时间：
2020-10
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影响因子：
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通讯作者：
M. Ben-Artzi;Jiequan Li
中科院分区：
文献类型：
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作者：
M. Ben-Artzi;Jiequan Li
本文讨论了非线性双曲型守恒律系统的紧有限体积格式中的一致性、稳定性和收敛性三个概念。这种处理运用了“平衡法则”的框架。这些定律表达了在不连续存在的情况下相关的物理守恒定律。有限体积近似采用了这一观点，本文可以认为属于这一范畴。首先证明了在非常温和的条件下，弱解确实是平衡律的解。这里考虑的方案允许计算每个网格单元的几个量（例如，斜率），一致性的概念必须扩展到这个框架。在此基础上，推广了经典的Lax和Wendroff收敛定理，建立了一个合适的收敛定理。最后，通过使用“Godunov相容性”的概念来证明极限函数是熵解，该概念可替代熵条件。
This paper addresses the three concepts of consistency, stability and convergence in the context of compact finite volume schemes for systems of nonlinear hyperbolic conservation laws. The treatment utilizes the framework of “balance laws”. Such laws express the relevant physical conservation laws in the presence of discontinuities. Finite volume approximations employ this viewpoint, and the present paper can be regarded as being in this category. It is first shown that under very mild conditions a weak solution is indeed a solution to the balance law. The schemes considered here allow the computation of several quantities per mesh cell (e.g., slopes) and the notion of consistency must be extended to this framework. Then a suitable convergence theorem is established, generalizing the classical convergence theorem of Lax and Wendroff. Finally, the limit functions are shown to be entropy solutions by using a notion of “Godunov compatibility”, which serves as a substitute to the entropy condition.