非线性双曲型方程的耦合技术. III. 厚界面的良平衡逼近
Coupling techniques for nonlinear hyperbolic equations. III. The well-balanced approximation of thick interfaces
AI总结：
针对可能共振界面的非线性双曲问题，提出基于厚界面模型的新耦合框架，并设计收敛到唯一熵解的良平衡有限体积格式。
摘要：
我们继续分析可能共振界面上非线性双曲问题的耦合。在本系列的前两部分中，我们引入了一个基于所谓薄界面模型的新耦合框架，该框架使用增广公式和界面位置的附加未知量；该框架的优点在于避免了对界面结构的显式建模。在本文中，我们继续研究增广公式，并引入一个新的耦合框架，该框架现在基于所谓的厚界面模型。对于一维空间变量中的标量非线性双曲方程，我们观察到柯西问题是适定的。然后，本文的主要成就是设计了一种新的适用于厚界面模型的良平衡有限体积格式，并证明了其收敛到唯一熵解（针对一大类非线性双曲方程）。由于可能存在共振界面，基于总变差估计的标准技术不适用，必须使用DiPerna的唯一性定理。遵循Coquel和LeFloch提出的方法，我们的证明依赖于耦合问题的离散熵不等式以及所提格式中离散熵耗散的估计。
英文摘要：
We continue our analysis of the coupling between nonlinear hyperbolic problems across possibly resonant interfaces. In the first two parts of this series, we introduced a new framework for coupling problems which is based on the so-called thin interface model and uses an augmented formulation and an additional unknown for the interface location; this framework has the advantage of avoiding any explicit modeling of the interface structure. In the present paper, we pursue our investigation of the augmented formulation and we introduce a new coupling framework which is now based on the so-called thick interface model. For scalar nonlinear hyperbolic equations in one space variable, we observe that the Cauchy problem is well-posed. Then, our main achievement in the present paper is the design of a new well-balanced finite volume scheme which is adapted to the thick interface model, together with a proof of its convergence toward the unique entropy solution (for a broad class of nonlinear hyperbolic equations). Due to the presence of a possibly resonant interface, the standard technique based on a total variation estimate does not apply, and DiPerna's uniqueness theorem must be used. Following a method proposed by Coquel and LeFloch, our proof relies on discrete entropy inequalities for the coupling problem and an estimate of the discrete entropy dissipation in the proposed scheme.
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